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  • Print publication year: 2014
  • Online publication date: December 2013

5 - A crash course on data analysis in asteroseismology

Summary

“Throughout human history, as our species has faced the frightening, terrorizing fact that we do not know who we are or where we are going in this ocean of chaos, it has been the authorities – the political, the religious, the educational authorities – who have attempted to comfort us by giving us order, rules, regulations, informing – forming in our minds – their view of reality.

To think for yourself you must question authority and learn how to put yourself in a state of vulnerable open-mindedness, chaotic, confused vulnerability to inform yourself.”

Timothy Leary in Sound Bites from the Counter Culture (1989)

Introduction

This chapter attempts to provide a summary of the course I gave during the XXII Canary Island Winter School of Astrophysics. In no way should this chapter be perceived as the final answer to a problem. I hope that this chapter can serve as a basis for students and fellow scientists to go beyond what is written here. As in many approaches that I have pursued, this work is a snapshot of where I am and hopefully a possible starting point from which one can expand to other paths not yet ventured.

This chapter starts with a short historical introduction on signal processing and statistics or how our forefathers started doing data analysis more than 200 years ago. The second part is related to the sampling and acquisition of continuous physical signals for subsequent analysis in a digital world.

Aigrain, S., Favata, F., and Gilmore, G. 2004. Characterising stellar micro-variability for planetary transit searches. A&A, 414, 1139–52.
Anderson, E.R., Duvall, T.L. Jr., and Jefferies, S. M. 1990. Modeling of solar oscillation power spectra. ApJ, 364(Dec.), 699.
Appourchaux, T. 2008. Bayesian approach for g-mode detection, or how to restrict our imagination. Astronomische Nachrichten, 329(June), 485.
Appourchaux, T., Andersen, B.N., Frohlich, C., Jimenez, A., Telljohann, U., and Wehrli, C. 1997. In-flight performance of the VIRGO Luminosity Oscillations Imager aboard SOHO. Sol. Phys., 170, 27–41.
Appourchaux, T., Gizon, L., and Rabello-Soares, M. C. 1998. The art of fitting p-mode spectra. I. Maximum Likelihood Estimation. A&AS, 132(Oct.), 107–19.
Appourchaux, T., Frohlich, C., Andersen, B.N., Berthomieu, G., Chaplin, W., Elsworth, Y., Finsterle, W., Gough, D.O., Hoeksema, J.T., Isaak, G.R., Kosovichev, A.G., Provost, J., Scherrer, P.H., Sekii, T., and Toutain, T. 2000. Observational upper limits for low-degree solar g modes. ApJ, 538, 401–14.
Appourchaux, T., Andersen, B., and Sekii, T. 2002. What have we learnt with the Luminosity Oscillations Imager over the past 6 years? Pages 47–50 of: C., Frohlich and A., Wilson (ed), From Solar Min to Max: Half a Solar Cycle with SOHO. ESA Special Publication, vol. 508.
Appourchaux, T., Berthomieu, G., Michel, E., Aerts, C., Ballot, J., Barban, C., Baudin, F., Boumier, P., de Ridder, J., Floquet, M., Garcia, R.A., Garrido, R., Goupil, M.-J., Lambert, P., Lochard, J., Neiner, C., Poretti, E., Provost, J., Roxburgh, I., Samadi, R., and Toutain, T. 2006a. Data analysis tools for the seismology programme. Page 377 of: M., Fridlung, A., Baglin, J., Lochard and L., Conroy (ed), The CoRoT Mission. ESA Publications Division, ESA Spec. Publ. 1306.
Appourchaux, T., Berthomieu, G., Michel, E., Ballot, J., Barban, C., Baudin, F., Boumier, P., de Ridder, J., Floquet, M., Garcia, R.A., Garrido, R., Goupil, M.-J., Lambert, P., Lochard, J., Mazumdar, A., Neiner, C., Poretti, E., Provost, J., Roxburgh, I., Samadi, R., and Toutain, T. 2006b. Evaluation of the scientific performances for the seismology programme. Page 429 of: M., Fridlung, A., Baglin, J., Lochard and L., Conroy (ed), The CoRoT Mission. ESA Publications Division, ESA Spec. Publ. 1306.
Appourchaux, T., Michel, E., Auvergne, M., Baglin, A., Toutain, T., Baudin, F., Benomar, O., Chaplin, W.J., Deheuvels, S.,Samadi, R.,Verner, G.A., Boumier, P., Garcia, R.A., Mosser, B., Hulot, J.-C., Ballot, J., Barban, C., Elsworth, Y., Jimenez-Reyes, S.J., Kjeldsen, H., Regulo, C., and Roxburgh, I. W. 2008. CoRoT sounds the stars: p-mode parameters of Sun-like oscillations on HD 49933. A&A, 488(Sept.), 705–14.
Appourchaux, T., Samadi, R., and Dupret, M.-A. 2009. On posterior probability and significance level: application to the power spectrum of HD 49 933 observed by CoRoT. A&A, 506 (Oct.), 1–5.
Ballot, J., Garcia, R.A., and Lambert, P. 2006. Rotation speed and stellar axis inclination from p modes: how CoRoT would see other suns. MNRAS, 369(July), 1281–1286.
Ballot, J., Appourchaux, T., Toutain, T., and Guittet, M. 2008. On deriving p-mode parameters for inclined solar-like stars. A&A, 486(Aug.), 867–75.
Barban, C., Deheuvels, S., Baudin, F., Appourchaux, T., Auvergne, M., Ballot, J., Boumier, P., Chaplin, W.J., Garcia, R.A., Gaulme, P., Michel, E., Mosser, B., Regulo, C., Roxburgh, I. W., Verner, G., Baglin, A., Catala, C., Samadi, R., Bruntt, H., Elsworth, Y., and Mathur, S. 2009. Solar-like oscillations in HD 181420: data analysis of 156 days of CoRoT data. A&A, 506(Oct.), 51–6.
Baudin, F., Barban, C., Belkacem, K., Hekker, S., Morel, T., Samadi, R., Benomar, O., Goupil, M., Carrier, F., Ballot, J., Deheuvels, S., De Ridder, J., Hatzes, A.P., Kallinger, T., and Weiss, W. W. 2011. Amplitudes and lifetimes of solar-like oscillations observed by CoRoT. Red-giant versus main-sequence stars. A&A, 535(May), A84.
Bayes, T. 1763. An Essay towards solving a Problem in the Doctrine of Chances. Philosophical Transactions of the Royal Society of London, 53, 370.
Bedding, T.R., Kjeldsen, H., Reetz, J., and Barbuy, B. 1996. Measuring stellar oscillations using equivalent widths of absorption lines. MNRAS, 280(June), 1155–61.
Belkacem, K., Samadi, R., Goupil, M.-J., Lefevre, L., Baudin, F., Deheuvels, S., Dupret, M.-A., Appourchaux, T., Scuflaire, R., Auvergne, M., Catala, C., Michel, E., Miglio, A., Montalban, J., Thoul, A., Talon, S., Baglin, A., and Noels, A. 2009. Solar-like oscillations in a massive star. Science, 324(June), 1540.
Belkacem, K., Dupret, M.A., and Noels, A. 2010. Solar-like oscillations in massive main-sequence stars. I. Asteroseismic signatures of the driving and damping regions. A&A, 510(Jan.), A6.
Belmonte, J.A., Chevreton, M., Mangeney, A., Praderie, F., Saint-Pe, O., Puget, P., Alvarez, M., and Roca Cortes, R. 1991. Rapid photometry of the Delta Scuti variable 63 Herculis and of the F 2V star HD 155543: Observations and analysis of the time series. A&A, 246(June), 71–83.
Benomar, O. 2010. Sismologie stellaire: Méthodes statistiques appliquées aux étoiles de type solaire. Ph.D. thesis, Universite Paris-Sud XI, Orsay, France.
Benomar, O., Baudin, F., Campante, T.L., Chaplin, W.J., Garcia, R.A., Gaulme, P., Toutain, T., Verner, G.A., Appourchaux, T., Ballot, J., Barban, C., Elsworth, Y., Mathur, S., Mosser, B., Regulo, C., Roxburgh, I.W., Auvergne, M., Baglin, A., Catala, C., Michel, E., and Samadi, R. 2009a. A fresh look at the seismic spectrum of HD49933: analysis of 180 days of CoRoT photometry. A&A, 507(Nov.), L13–L16.
Benomar, O., Appourchaux, T., and Baudin, F. 2009b. The solar-like oscillations of HD 49933: a Bayesian approach. A&A, 506(Oct.), 15–32.
Berger, J. O. 1997. Could Fisher, Jeffreys and Neyman have agreed upon testing?Statistical Science, 18, 1–32.
Berger, J.O., and Sellke, T. 1987. Testing of a point null hypothesis: The irreconcilability of significance levels and evidence. Journal of the American Statistical Association, 82(397), 112–39.
Berger, J.O., Boukai, B., and Wang, Y. 1997. Unified frequentist and Bayesian testing of a precise hypothesis. Statistical Science, 12(3), 133–60.
Bernoulli, J. 1713. Ars conjectandi, opus posthumum. Accedit Tractatus de seriebus infinitis, et epistola gallice scripta de ludo pilae reticularis. Thurneysen Brothers, Basel, Switzerland.
Bretthorst, L. G. 1988. Bayesian Spectrum Analysis and Parameter Estimation. Springer-Verlag, Berlin, available electronically at bayes.wustl.edu/glb/book.pdf.
Bretthorst, G. L. 2001a. Generalizing the Lomb-Scargle periodogram. Pages 241–245 of: A., Mohammad-Djafari (ed), Bayesian Inference and Maximum Entropy Methods in Science and Engineering. American Institute of Physics Conference Series, vol. 568.
Bretthorst, G. L. 2001b. Generalizing the Lomb-Scargle periodogram: the nonsinusoidal case. Pages 246–251 of: A., Mohammad-Djafari (ed), Bayesian Inference and Maximum Entropy Methods in Science and Engineering. American Institute of Physics Conference Series, vol. 568.
Brewer, B.J., Bedding, T.R., Kjeldsen, H., and Stello, D. 2007. Bayesian inference from observations of solar-like oscillations. ApJ, 654(Jan.), 551–7.
Broomhall, A.-M., Chaplin, W.J., Elsworth, Y., Appourchaux, T., and New, R. 2010. A comparison of frequentist and Bayesian inference: searching for low-frequency p modes and g modes in Sun-as-a-star data. MNRAS, 406(Aug.), 767–81.
Chaplin, W.J., Elsworth, Y., Howe, R., Isaak, G.R., McLeod, C.P., Miller, B.A., van der Raay, H.B., Wheeler, S.J., and New, R. 1996. BiSON Performance. Sol. Phys., 168(Sept.), 1–18.
Chaplin, W.J., Appourchaux, T., Elsworth, Y., Isaak, G.R., Miller, B.A., and New, R. 2004. On comparing estimates of low-l solar p-mode frequencies from Sun-as-a-star and resolved observations. A&A, 424(Sept.), 713–17.
Christensen-Dalsgaard, J. 1989. The effect of rotation on whole-disc Doppler observations of solar oscillations. MNRAS, 239(Aug.), 977–94.
Christensen-Dalsgaard, J. 2004. An Overview of helio- and asteroseismology. Page 1 of: D., Danesy (ed), SOHO 14 Helio- and Asteroseismology: Towards a Golden Future. ESA Special Publication, vol. 559.
Christensen-Dalsgaard, J., and Gough, D. O. 1982. On the interpretation of five-minute oscillations in solar spectrum line shifts. MNRAS, 198(Jan.), 141–71.
Christensen-Dalsgaard, J., Carpenter, K.G., Schrijver, C.J., Karovska, M., and the Si Team. 2011. The stellar imager (SI): a mission to resolve stellar surfaces, interiors, and magnetic activity. Journal of Physics Conference Series, 271(1), 012085.
Cooley, J.W., and Tuckey, J. W. 1965. An algorithm for the machine calculation of complex Fourier series. Math. Comp., 19, 297–301.
Cramer, H. 1946. Mathematical Methods of Statistics. Princeton University Press, New Jersey.
Cuypers, J. 1987. Medelingen van de Koninklijke Academie voor Wetenschappen, Letteren en Schone Kunsten van Belgi'e, Jg. 49 Nr. 3 (Brussel: Paleis der Academieen), 21.
De Finetti, B. 1937. La prevision: ses lois logiques, ses sources subjectives. Annales de l'Institut Henri Poincare, 7, 1.
De Moivre, A. 1718. The doctrine of chances of a method of calculating the probability of events in play. W. Pearson, London, United Kingdom.
Deheuvels, S., Bruntt, H., Michel, E., Barban, C., Verner, G., Regulo, C., Mosser, B., Mathur, S., Gaulme, P., Garcia, R.A., Boumier, P., Appourchaux, T., Samadi, R., Catala, C., Baudin, F., Baglin, A., Auvergne, M., Roxburgh, I.W., and Perez Hernandez, F. 2010. Seismic and spectroscopic characterization of the solar-like pulsating CoRoT target HD 49385. A&A, 515(June), A87.
Duvall, T.L. Jr., and Harvey, J. W. 1986. Solar Doppler shifts: sources of continuous spectra. Page 105 of: Osaki, Y., and Shibahashi, H. (eds), Seismology of the Sun and the Distant Stars. Lecture Notes in Physics, Springer Verlag, Berlin.
Duvall, T. L. Jr., Jefferies, S.M., Harvey, J.W., Osaki, Y., and Pomerantz, M. A. 1993. Asymmetries of solar oscillation line profiles. ApJ, 410(June), 829–36.
Fisher, R. A. 1912. On an absolute criterion for fitting frequency curves. Messenger of Mathematics, 41, 155–60.
Fisher, R. A. 1925. Theory of statistical estimation. Proc. Cambridge Philos. Soc., 22, 155–60.
Fleischmann, M., and Pons, S. 1989. Electrochemically induced nuclear fusion of deuterium. Journal of Electroanalytical Chemistry and Interfacial Electrochemistry, 261, 301–8.
Fourier, J. 1822. Theorieanalytiquedelachaleur. Firmin-Didot, Paris, France.
Frieden, B. R. 1983. Probability, Statistical Optics, and Data Testing: A Problem Solving Approach. Springer-Verlag, Berlin.
Frohlich, C., Andersen, B.N., Appourchaux, T., Berthomieu, G., Crommelynck, D.A., Domingo, V., Fichot, A., Finsterle, W., Gomez, M.F., Gough, D., Jimenez, A., Leifsen, T., Lombaerts, M., Pap, J.M., Provost, J., Cortes, T.R., Romero, J., Roth, H., Sekii, T., Telljohann, U., Toutain, T., and Wehrli, C. 1997. First results from VIRGO, the experiment for helioseis-mology and solar irradiance monitoring on SOHO. Sol. Phys., 170, 1–25.
Gabriel, M. 1993. The probability-density function of solar p modes and the location of the excitation mechanism. A&A, 274(July), 931.
Gabriel, M. 1994. The probability-density function of a Fourier line. A&A, 287(July), 685–691.
Garcia, R.A., Regulo, C., Samadi, R., Ballot, J., Barban, C., Benomar, O., Chaplin, W.J., Gaulme, P., Appourchaux, T., Mathur, S., Mosser, B., Toutain, T., Verner, G.A., Auvergne, M., Baglin, A., Baudin, F., Boumier, P., Bruntt, H., Catala, C., Deheuvels, S., Elsworth, Y., Jimenez-Reyes, S.J., Michel, E., Perez Hernandez, F., Roxburgh, I.W., and Salabert, D. 2009. Solar-like oscillations with low amplitude in the CoRoT target HD 181906. A&A, 506(Oct.), 41–50.
Gaulme, P., Appourchaux, T., and Boumier, P. 2009. Mode width fitting with a simple Bayesian approach. Application to CoRoT targets HD 181420 and HD 49933. A&A, 506(Oct.), 7–14.
Gauss, C. F. 1809. Theoria motus corporum coelestium in sectionibus conicis solem ambientum. Konigliche Gesellschaft der Wissenschaften, Gottingen, Germany.
Gauss, C. F. 1866. Nachlass: Theoria interpolationis methodo nova tractata: Werke band 3, 265-327. Konigliche Gesellschaft der Wissenschaften, Gottingen, Germany.
Gautschy, A., and Saio, H. 1995. Stellar pulsations across the HR diagram: part 1. ARA&A, 33, 75–114.
Gautschy, A., and Saio, H. 1996. Stellar pulsations across the HR diagram: part 2. ARA&A, 34, 551–606.
Gelly, B., Lazrek, M., Grec, G., Ayad, A., Schmider, F.X., Renaud, C., Salabert, D., and Fossat, E. 2002. Solar p-modes from 1979 days of the GOLF experiment. A&A, 394(Oct.), 285.
Gelman, A., and Meng, X.-L. 1998. Simulating normalizing constants: from importance sampling to bridge sampling to path sampling. Statistical Science, 13(2)(Oct.), 163–185.
Gibbs, J. W. 1898. Fourier series. Nature, 59, 200.
Gibbs, J. W. 1899. Fourier series. Nature, 59, 606.
Gizon, L., and Solanki, S. K. 2003. Determining the inclination of the rotation axis of a sun-like star. ApJ, 589(June), 1009–19.
Goldstine, H. H. 1977. A History of Numerical Analysis from the 16th through the 19th Century. Springer, New York-Heidelberg-Berlin.
Gott, J. R. III 1994. Future Prospects Discussed. Nature, 368, 106–8.
Gregory, P. C. 2005. Bayesian Logical Data Analysis for the Physical Sciences: A Comparative Approach with “Mathematica” Support. Cambridge University Press, Cambridge.
Gruberbauer, M., Kallinger, T., Weiss, W.W., and Guenther, D. B. 2009. On the detection of Lorentzian profiles in a power spectrum: a Bayesian approach using ignorance priors. A&A, 506(Nov.), 1043–53.
Handberg, R., and Campante, T. L. 2011. Bayesian peak-bagging of solar-like oscillators using MCMC: a comprehensive guide. A&A, 527(Mar.), A56.
Harvey, J. 1985. High-resolution helioseismology. Pages 199–208 of: E., Rolfe, and B., Battrick (eds), Future missions in solar, heliospheric and space plasma physics, ESA SP-235. ESA Publications Division, Noordwijk, The Netherlands.
Harvey, J.W., Duvall, T. L. Jr., Jefferies, S.M., and Pomerantz, M. A. 1993. Chromospheric oscillations and the background spectrum. ASP Conference Series, 42, 111.
Hastings, W. K. 1970. Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57(1)(July), 97.
Heideman, M.T., Johnson, D.H., and Sidney Burrus, C. 1985. Gauss and the history of the fast Fourier transform. Archive for History of Exact Sciences, 34, 265–77.
Hill, F., Stark, P.B., Stebbins, R.T., Anderson, E.R., Antia, H.M., Brown, T.M., Duvall, T. L. Jr., Haber, D.A., Harvey, J.W., Hathaway, D.H., Howe, R., Hubbard, R., Jones, H.P., Kennedy, J.R., Korzennik, S.G., Kosovichev, A.G., Leibacher, J.W., Libbrecht, K.G., Pintar, J.A., Rhodes, E. J. Jr., Schou, J.Thompson, M.J., Tomczyk, S., Toner, C.G., Toussaint, R., and Williams, W. E. 1996. The Solar Acoustic Spectrum and Eigenmode. Science, 272, 1292.
Hogbom, J. A. 1974. Aperture Synthesis with a Non-Regular Distribution of Interferometer Baselines. A&AS, 15(June), 417–26.
Houdek, G., Balmforth, N.J., Christensen-Dalsgaard, J., and Gough, D. O. 1999. Amplitudes of stochastically excited oscillations in main-sequence stars. A&A, 351(Nov.), 582.
Jaynes, E. T. 1987. Bayesian spectrum and chirp analysis. Page 1 of: C. R., Smith and G. J., Erickson (ed), Maximum entropy and Bayesian spectral analysis and estimation problems. D.Reidel, Dordrecht, the Netherlands.
Jaynes, E. T. 2009. Probability Theory: The Logic of Science. 6th edition. Larry, Brethorst (ed). Cambridge University Press, Cambridge, United Kingdom.
Jeffreys, H. 1939. Theory of Probability. Oxford Universitty Press, Oxford, United Kingdom.
Jeffreys, H. 1946. An invariant form for the prior probability in estimation problems. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 186, 155–160.
Keshner, M. S. 1982. 1/f noise. IEEE Proceedings, 70, 212–18.
Khinchin, A. Y. 1934. Korrelationstheorie der stationaren stochastichen Prozesse. Mathematische Annalen, 109, 604.
Koen, C. 1999. The analysis of indexed astronomical time series. V. Fitting sinusoids to highspeed photometry. MNRAS, 309(Nov.), 769–802.
Komm, R.W., Gu, Y., Hill, F., Stark, P.B., and Fodor, I. K. 1999. Multitaper spectral analysis and wavelet denoising applied to helioseismic data. ApJ, 519, 407–21.
Kosovichev, A.G., Schou, J., Scherrer, P.H., Bogart, R.S., Bush, R.I., Hoeksema, J.T., Aloise, J., Bacon, L., Burnette, A., de Forest, C., Giles, P.M., Leibrand, K., Nigam, R., Rubin, M., Scott, K., Williams, S.D., Basu, S., Christensen-Dalsgaard, J., Dappen, W., Rhodes, E.J., Duvall, T.L., Howe, R., Thompson, M.J., Gough, D.O., Sekii, T., Toomre, J., Tarbell, T.D., Title, A.M., Mathur, D., Morrison, M., Saba, J. L. R., Wolfson, C.J., Zayer, I., and Milford, P. N. 1997. Structure and rotation of the solar interior: initial results from the MDI medium-l program. Sol. Phys., 170, 43–61.
Kumar, P., Franklin, J., and Goldreich, P. 1988. Distribution functions for the time-averaged energies of stochastically excited solar p modes. ApJ, 328(May), 879.
Laplace, M. 1774. Memoire sur la probabilité des causes par les événements. Mémoires de Mathématique et de Physique, T6, 621–56.
Legendre, A.-M. 1805. Nouvelles méthodes pour la determination des orbites des cometes. Firmin Didot, Paris, France.
Libbrecht, K. G. 1992. On the ultimate accuracy of solar oscillation frequency measurements. ApJ, 387(Mar.), 712.
Mathur, S., Handberg, R., Campante, T.L., Garcia, R.A., Appourchaux, T., Bedding, T.R., Mosser, B., Chaplin, W.J., Ballot, J., Benomar, O., Bonanno, A., Corsaro, E., Gaulme, P., Hekker, S., Regulo, C., Salabert, D., Verner, G., White, T.R., Brandão, I.M., Creevey, O.L., Dogan, G., Elsworth, Y., Huber, D., Hale, S.J., Houdek, G., Karoff, C., Metcalfe, T.S., Molenda-Zakowicz, J., Monteiro, M. J. P. F. G., Thompson, M.J., Christensen-Dalsgaard, J., Gilliland, R.L., Kawaler, S.D., Kjeldsen, H., Quintana, E.V., Sanderfer, D.T., and Seader, S. E. 2011. Solar-like oscillations in KIC11395018 and KIC11234888 from 8 months of Kepler data. ArXiv e-prints.
Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., and Teller, E. 1953. Equation of state calculations by fast computing machines. Journal of Chemical Physics, 21(6) (July), 1087.
Mosser, B., Michel, E., Appourchaux, T., Barban, C., Baudin, F., Boumier, P., Bruntt, H., Catala, C., Deheuvels, S., Garcia, R.A., Gaulme, P., Regulo, C., Roxburgh, I., Samadi, R., Verner, G., Auvergne, M., Baglin, A., Ballot, J., Benomar, O., and Mathur, S. 2009. The CoRoT target HD 175726: an active star with weak solar-like oscillations. A&A, 506(Oct.), 33–40.
Neyman, J., and Pearson, E. 1933. On the problem of the most efficient tests of statistical hypotheses. Philosophical Transactions of the Royal Society of London. Series A, 231(July), 289.
Nigam, R., and Kosovichev, A. G. 1998. Measuring the Sun's eigenfrequencies from velocity and intensity helioseismic spectra: asymmetrical line profile-fitting formula. ApJ, 505(Sept.), L51.
Nigam, R., Kosovichev, A.G., Scherrer, P.H., and Schou, J. 1998. Asymmetry in velocity and intensity helioseismic spectra: a solution to a long-standing puzzle. ApJ, 495(Mar.), L115.
Nyquist, H. 1924. Certain factors affecting telegraph speed. Bell Syst. Tech. Journal, 3, 324–52.
Parseval des Chênes, M.-A. 1806. Mémoire sur les séries et sur l'intégration complete d'une équation aux differences partielles lineaire du second ordre, a coefficients constants. Mémoires présentés a l'Institut des Sciences, Lettres et Arts, par divers savans, et lus dans ses assemblees. Sciences, mathématiques et physiques, 1, 638.
Peligrad, M., and Wu, W. B. 2010. Central limit theorem for Fourier transforms of stationary processes. The Annals of Probability, 38, 2009.
Poincare, H. 1914. La Science et l'Hypothèse. Editions Flammarion, Paris, France.
Poretti, E., Michel, E., Garrido, R., Lefevre, L., Mantegazza, L., Rainer, M., Rodríguez, E., Uytterhoeven, K., Amado, P.J., Martin-Ruiz, S., Moya, A., Niemczura, E., Suarez, J.C., Zima, W., Baglin, A., Auvergne, M., Baudin, F., Catala, C., Samadi, R., Alvarez, M., Mathias, P., Paparo, M., Papics, P., and Plachy, E. 2009. HD 50844: a new look at delta Scuti stars from CoRoT space photometry. A&A, 506(Oct.), 85–93.
Press, W.H., and Rybicki, G. B. 1989. Fast algorithm for spectral analysis of unevenly sampled data. ApJ, 338(Mar.), 277–80.
Rao, C.R., 1945. Information and the accuracy attainable in the estimation of statistical parameters. Bulletin of the Calcutta Mathematical Society, 37, 81–9.
Ritzwoller, M.H., and Lavely, E. M. 1991. A unified approach to the helioseismic forward and inverse problems of differential rotation. ApJ, 369(Mar.), 557.
Roberts, D.H., Lehar, J., and Dreher, J. W. 1987. Time Series Analysis with clean. Part one: derivation of a spectrum. AJ, 93(Apr.), 968–89.
Samadi, R., and Goupil, M.-J. 2001. Excitation of stellar p-modes by turbulent convection. I. Theoretical formulation. A&A, 370, 136–46.
Scargle, J. D. 1982. Studies in astronomical time series analysis. II. Statistical aspects of spectral analysis of unevenly spaced data. ApJ, 263(Dec.), 835–53.
Schou, J. 1992. On the analysis of helioseismic data. Ph.D. thesis, Arhus Universitet, Denmark.
Schou, J., and Brown, T. M. 1994. Generation of artificial helioseismic time-series. A&AS, 107(Nov.), 541.
Schuster, A. 1897. On lunar and solar periodicities of earthquakes. Proc. Royal Soc. of London, 61, 455.
Scott, J.G., and Berger, J. O. 2010. Bayes and empirical-Bayes multiplicity adjustment in the variable-selection problem. The Annals of Statistics, 38(5), 2587–619.
Sellke, T., Bayarri, M.J., and Berger, J. 2001. Calibration of p-values for testing precise null hypotheses. The American Statistician, 55(Mar.), 62–71.
Severino, G., Magrí, M., Oliviero, M., Straus, T., and Jefferies, S. M. 2001. The Solar intensity-velocity cross spectrum: a powerful diagnostic for helioseismology. ApJ, 561(Nov.), 444–9.
Shannon, C. E. 1949. Communication in the presence of noise. Proceedings of the IRE, 37, 10–21.
Slepian, D. 1978. Prolate spheroidal wave functions, Fourier analysis, and uncertainty. V. The discrete case. Bell System Technical Journal, 57, 1371–430.
Stahn, T., and Gizon, L. 2008. Fourier analysis of gapped time series: improved estimates of solar and stellar oscillation parameters. Sol. Phys., 251(Sept.), 31–52.
Sturrock, P.A., and Scargle, J. D. 2009. A Bayesian assessment of p-values for significance estimation of power spectra and an alternative procedure, with application to solar neutrino data. ApJ, 706(Nov.), 393–8.
Tanvir, N.R., Hendry, M.A., Watkins, A., Kanbur, S.M., Berdnikov, L.N., and Ngeow, C. C. 2005. Determination of Cepheid parameters by light-curve template fitting. MNRAS, 363(Nov.), 749–62.
Thomson, D. J. 1982. Spectrum estimation and harmonic analysis. Pages 1055–1096 of: Proc. IEEE, vol. 70.
Toutain, T., and Appourchaux, T. 1994. Maximum likelihood estimators: an application to the estimation of the precision of helioseismic measurements. A&A, 289(Sept.), 649.
Toutain, T., and Frohlich, C. 1992. Characteristics of solar p-modes: results from the IPHIR experiment. A&A, 257(Apr.), 287–97.
Toutain, T., and Gouttebroze, P. 1993. Visibility of solar p-modes. A&A, 268(Feb.), 309–18.
Toutain, T., Appourchaux, T., Baudin, F., Frohlich, C., Gabriel, A.H., Scherrer, P., Andersen, B.N., Bogart, R., Bush, R., Finsterle, W., Garciía, R.A., Grec, G., Henney, C.J., Hoeksema, J. T., Jimenez, A., Kosovichev, A., Roca Cortés, T., Turck-Chiěze, S., Ulrich, R., and Wehrli, C. 1997. Tri-phonic helioseismology: comparison of solar p modes observed by the hélioséismology instruments aboard SOHO. Sol. Phys., 175(Oct.), 311–28.
Wehlau, W., and Leung, K.-C. 1964. The multiple periodicity of Delta Delphini. ApJ, 139(Apr.), 843.
Wiener, N. 1930. Generalized harmonic analysis. Acta Mathematica, 55, 117.
Wilks, S. S. 1938. The large-sample distribution of the likelihood ratio for testing com?posite hypotheses. Annals of Mathematical Statistics, 9(July), 60.
Yoachim, P., McCommas, L.P., Dalcanton, J.J., and Williams, B. F. 2009. A panoply of Cepheid light curve templates. AJ, 137(June), 4697–706.