Annals of Applied Statistics

Estimating a common period for a set of irregularly sampled functions with applications to periodic variable star data

James P. Long, Eric C. Chi, and Richard G. Baraniuk

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We consider the problem of estimating a common period for a set of functions sampled at irregular intervals. The motivating problem arises in astronomy, where the functions represent a star’s observed brightness over time through different photometric filters. While current methods perform well when the brightness is sampled densely enough in at least one filter, they break down when no brightness function is densely sampled. In this paper we introduce two new methods for period estimation in this important latter case. The first, multiband generalized Lomb–Scargle (MGLS), extends the frequently used Lomb–Scargle method to naïvely combine information across filters. The second, penalized generalized Lomb–Scargle (PGLS), builds on MGLS by more intelligently borrowing strength across filters. Specifically, we incorporate constraints on the phases and amplitudes across the different functions using a nonconvex penalized likelihood function. We develop a fast algorithm to optimize the penalized likelihood that combines block coordinate descent with the majorization–minimization (MM) principle. We test and validate our methods on synthetic and real astronomy data. Both PGLS and MGLS improve period estimation accuracy over current methods based on using a single function; moreover, PGLS outperforms MGLS and other leading methods when the functions are sparsely sampled.

Article information

Ann. Appl. Stat., Volume 10, Number 1 (2016), 165-197.

Received: December 2014
Revised: July 2015
First available in Project Euclid: 25 March 2016

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Astrostatistics penalized likelihood period estimation functional data MM algorithm block coordinate descent


Long, James P.; Chi, Eric C.; Baraniuk, Richard G. Estimating a common period for a set of irregularly sampled functions with applications to periodic variable star data. Ann. Appl. Stat. 10 (2016), no. 1, 165--197. doi:10.1214/15-AOAS885.

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  • Allen, G. I. (2013). Sparse and functional principal components analysis. Available at arXiv:1309.2895.
  • Allen, G. I., Grosenick, L. and Taylor, J. (2014). A generalized least-square matrix decomposition. J. Amer. Statist. Assoc. 109 145–159.
  • Becker, M. P., Yang, I. and Lange, K. (1997). EM algorithms without missing data. Stat. Methods Med. Res. 6 38–54.
  • Böhning, D. and Lindsay, B. G. (1988). Monotonicity of quadratic-approximation algorithms. Ann. Inst. Statist. Math. 40 641–663.
  • Chung, F. R. K. (1997). Spectral Graph Theory. CBMS Regional Conference Series in Mathematics 92. Amer. Math. Soc., Providence, RI.
  • Debosscher, J., Sarro, L., López, M., Deleuil, M., Aerts, C., Auvergne, M., Baglin, A., Baudin, F., Chadid, M., Charpinet, S. et al. (2009). Automated supervised classification of variable stars in the CoRoT programme. Astron. Astrophys. 506 519.
  • Dubath, P., Rimoldini, L., Süveges, M., Blomme, J., López, M., Sarro, L. M., De Ridder, J., Cuypers, J., Guy, L., Lecoeur, I. et al. (2011). Random forest automated supervised classification of Hipparcos periodic variable stars. Mon. Not. R. Astron. Soc. 414 2602–2617.
  • Freedman, W. L. (1988). New Cepheid distances to nearby galaxies based on BVRI CCD photometry. I. IC 1613. Astrophys. J. 326 691–709.
  • Friedman, J. H. (1984). A variable span smoother. Technical report, Stanford Univ., Stanford, CA.
  • Gelfand, A. E. and Vounatsou, P. (2003). Proper multivariate conditional autoregressive models for spatial data analysis. Biostatistics 4 11–15.
  • Graham, M. J., Drake, A. J., Djorgovski, S. G., Mahabal, A. A., Donalek, C., Duan, V. and Maker, A. (2013). A comparison of period finding algorithms. Mon. Not. R. Astron. Soc. 434 3423–3444.
  • Hastie, T., Tibshirani, R. and Friedman, J. (2009). The Elements of Statistical Learning: Data Mining, Inference, and Prediction, 2nd ed. Springer, New York.
  • Hoerl, A. E. and Kennard, R. W. (1970). Ridge regression: Biased estimation for nonorthogonal problems. Technometrics 12 55–67.
  • Huang, J. Z., Shen, H. and Buja, A. (2009). The analysis of two-way functional data using two-way regularized singular value decompositions. J. Amer. Statist. Assoc. 104 1609–1620.
  • Ivezić, Ž., Smith, J. A., Miknaitis, G., Lin, H., Tucker, D., Lupton, R. H., Gunn, J. E., Knapp, G. R., Strauss, M. A., Sesar, B. et al. (2007). Sloan digital sky survey standard star catalog for stripe 82: The dawn of industrial 1% optical photometry. Astron. J. 134 973.
  • Keller, J. P., Olives, C., Kim, S. Y., Sheppard, L., Sampson, P. D., Szpiro, A. A., Oron, A. P., Lindström, J., Vedal, S. and Kaufman, J. D. (2015). A unified spatiotemporal modeling approach for predicting concentrations of multiple air pollutants in the Multi-Ethnic Study of Atherosclerosis and Air Pollution. Environ. Health Perspect. 123 301–309.
  • Lange, K., Hunter, D. R. and Yang, I. (2000). Optimization transfer using surrogate objective functions. J. Comput. Graph. Statist. 9 1–59.
  • Li, C. and Li, H. (2008). Network-constrained regularization and variable selection for analysis of genomic data. Bioinformatics 24 1175–1182.
  • Lomb, N. R. (1976). Least-squares frequency analysis of unequally spaced data. Astrophys. Space Sci. 39 447–462.
  • Long, J. P., Chi, E. C. and Baraniuk, R. G. (2014). Estimating a common period for a set of irregularly sampled functions with applications to periodic variable star data. Preprint. Available at arXiv:1412.6520.
  • Meyer, R. R. (1976). Sufficient conditions for the convergence of monotonic mathematical programming algorithms. J. Comput. System Sci. 12 108–121.
  • Percy, J. R. (2007). Understanding Variable Stars. Cambridge Univ. Press, Cambridge.
  • Ramsay, J. O. (2004). Functional Data Analysis. In Encyclopedia of Statistical Sciences. Wiley, New York.
  • Rañola, J. M., Novembre, J. and Lange, K. (2014). Fast spatial ancestry via flexible allele frequency surfaces. Bioinformatics 30 2915–2922.
  • Reimann, J. D. (1994). Frequency estimation using unequally-spaced astronomical data. Ph.D. thesis, Univ. California.
  • Richards, J. W., Starr, D. L., Butler, N. R., Bloom, J. S., Brewer, J. M., Crellin-Quick, A., Higgins, J., Kennedy, R. and Rischard, M. (2011). On machine-learned classification of variable stars with sparse and noisy time-series data. Astrophys. J. 733 10.
  • Riess, A. G., Macri, L., Casertano, S., Lampeitl, H., Ferguson, H. C., Filippenko, A. V., Jha, S. W., Li, W. and Chornock, R. (2011). A 3% solution: Determination of the Hubble constant with the Hubble space telescope and wide field camera 3. Astrophys. J. 730 119.
  • Scargle, J. D. (1982). Studies in astronomical time series analysis. II-Statistical aspects of spectral analysis of unevenly spaced data. Astrophys. J. 263 835–853.
  • Schlafly, E. F., Finkbeiner, D. P., Jurić, M., Magnier, E. A., Burgett, W. S., Chambers, K. C., Grav, T., Hodapp, K. W., Kaiser, N., Kudritzki, R.-P. et al. (2012). Photometric calibration of the first 1.5 years of the Pan-STARRS1 survey. Astrophys. J. 756 158.
  • Sesar, B., Ivezić, Ž., Lupton, R. H., Jurić, M., Gunn, J. E., Knapp, G. R., De Lee, N., Smith, J. A., Miknaitis, G., Lin, H. et al. (2007). Exploring the variable sky with the sloan digital sky survey. Astron. J. 134 2236.
  • Sesar, B., Ivezić, Ž., Grammer, S. H., Morgan, D. P., Becker, A. C., Jurić, M., De Lee, N., Annis, J., Beers, T. C., Fan, X. et al. (2010). Light curve templates and galactic distribution of RR Lyrae stars from Sloan Digital Sky Survey Stripe 82. Astrophys. J. 708 717.
  • Shappee, B. J. and Stanek, K. Z. (2011). A new Cepheid distance to the giant spiral M101 based on image subtraction of Hubble Space Telescope/Advanced Camera for Surveys observations. Astrophys. J. 733 124.
  • Stetson, P. B. (1996). On the automatic determination of light-curve parameters for Cepheid variables. Publ. Astron. Soc. Pac. 108 851–876.
  • Suveges, M., Sesar, B., Varadi, M., Mowlavi, N., Becker, A. C., Ivezić, Ž. Beck, M., Nienartowicz, K., Rimoldini, L., Dubath, P. et al. (2012). Search for high-amplitude Scuti and RR Lyrae stars in Sloan Digital Sky Survey Stripe 82 using principal component analysis. Mon. Not. R. Astron. Soc. 424 2528–2550.
  • Tian, T. S., Huang, J. Z., Shen, H. and Li, Z. (2012). A two-way regularization method for MEG source reconstruction. Ann. Appl. Stat. 6 1021–1046.
  • Torabi, M. and Rosychuk, R. J. (2010). Spatio-temporal modelling of disease mapping of rates. Canad. J. Statist. 38 698–715.
  • Udalski, A., Szymanski, M. K., Soszynski, I. and Poleski, R. (2008). The Optical Gravitational Lensing Experiment. Final reductions of the OGLE-III data. Acta Astron. 58 69–87.
  • VanderPlas, J. T. and Ivezić, Ž. (2015). Periodograms for multiband astronomical time series. Preprint. Available at arXiv:1502.01344.
  • Watkins, L. L., Evans, N. W., Belokurov, V., Smith, M. C., Hewett, P. C., Bramich, D. M., Gilmore, G. F., Irwin, M. J., Vidrih, S., Zucker, D. B. et al. (2009). Substructure revealed by RR Lyraes in SDSS Stripe 82. Mon. Not. R. Astron. Soc. 398 1757–1770.
  • Welch, D. L. and Stetson, P. B. (1993). Robust variable star detection techniques suitable for automated searches—New results for NGC 1866. Astron. J. 105 1813–1821.
  • Zechmeister, M. and Kürster, M. (2009). The generalised Lomb–Scargle periodogram. A new formalism for the floating-mean and Keplerian periodograms. Astron. Astrophys. 496 577–584.