The Annals of Applied Statistics

Detecting abrupt changes in the spectra of high-energy astrophysical sources

Raymond K. W. Wong, Vinay L. Kashyap, Thomas C. M. Lee, and David A. van Dyk

Full-text: Open access

Abstract

Variable-intensity astronomical sources are the result of complex and often extreme physical processes. Abrupt changes in source intensity are typically accompanied by equally sudden spectral shifts, that is, sudden changes in the wavelength distribution of the emission. This article develops a method for modeling photon counts collected from observation of such sources. We embed change points into a marked Poisson process, where photon wavelengths are regarded as marks and both the Poisson intensity parameter and the distribution of the marks are allowed to change. To the best of our knowledge, this is the first effort to embed change points into a marked Poisson process. Between the change points, the spectrum is modeled nonparametrically using a mixture of a smooth radial basis expansion and a number of local deviations from the smooth term representing spectral emission lines. Because the model is over-parameterized, we employ an $\ell_{1}$ penalty. The tuning parameter in the penalty and the number of change points are determined via the minimum description length principle. Our method is validated via a series of simulation studies and its practical utility is illustrated in the analysis of the ultra-fast rotating yellow giant star known as FK Com.

Article information

Source
Ann. Appl. Stat., Volume 10, Number 2 (2016), 1107-1134.

Dates
Received: August 2015
Revised: December 2015
First available in Project Euclid: 22 July 2016

Permanent link to this document
https://projecteuclid.org/euclid.aoas/1469199907

Digital Object Identifier
doi:10.1214/16-AOAS933

Mathematical Reviews number (MathSciNet)
MR3528374

Zentralblatt MATH identifier
06625683

Keywords
Astronomy change point estimation FK Comae Berenices marked poisson process minimum description length principle semi-parametric modeling stars stellar coronae stellar flare X-ray astronomy

Citation

Wong, Raymond K. W.; Kashyap, Vinay L.; Lee, Thomas C. M.; van Dyk, David A. Detecting abrupt changes in the spectra of high-energy astrophysical sources. Ann. Appl. Stat. 10 (2016), no. 2, 1107--1134. doi:10.1214/16-AOAS933. https://projecteuclid.org/euclid.aoas/1469199907


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References

  • Akman, V. E. and Raftery, A. E. (1986). Asymptotic inference for a change-point Poisson process. Ann. Statist. 14 1583–1590.
  • Aue, A. and Lee, T. C. M. (2011). On image segmentation using information theoretic criteria. Ann. Statist. 39 2912–2935.
  • Aue, A., Cheung, R. C. Y., Lee, T. C. M. and Zhong, M. (2014). Segmented model selection in quantile regression using the minimum description length principle. J. Amer. Statist. Assoc. 109 1241–1256.
  • Bopp, B. W. and Stencel, R. E. (1981). The FK comae stars. Astrophys. J. 247 L131–L134.
  • Buhmann, M. D. (2003). Radial Basis Functions: Theory and Implementations. Cambridge Monographs on Applied and Computational Mathematics 12. Cambridge Univ. Press, Cambridge.
  • Carlin, B. P., Gelfand, A. E. and Smith, A. F. (1992). Hierarchical Bayesian analysis of changepoint problems. Appl. Stat. 41 389–405.
  • Chan, H. P. and Zhang, N. R. (2007). Scan statistics with weighted observations. J. Amer. Statist. Assoc. 102 595–602.
  • Chen, J. and Chen, Z. (2008). Extended Bayesian information criteria for model selection with large model spaces. Biometrika 95 759–771.
  • Chib, S. (1998). Estimation and comparison of multiple change-point models. J. Econometrics 86 221–241.
  • Cohen, O., Drake, J. J., Kashyap, V. L., Korhonen, H., Elstner, D. and Gombosi, T. I. (2010). Magnetic structure of rapidly rotating FK comae-type coronae. Astrophys. J. 719 299–306.
  • Davis, R. A. and Yau, C. Y. (2013). Consistency of minimum description length model selection for piecewise stationary time series models. Electron. J. Stat. 7 381–411.
  • Drake, J. J., Chung, S. M., Kashyap, V., Korhonen, H., Van Ballegooijen, A. and Elstner, D. (2008). X-ray spectroscopic signatures of the extended corona of FK comae. Astrophys. J. 679 1522–1530.
  • Elstner, D. and Korhonen, H. (2005). Flip-flop phenomenon: Observations and theory. Preprint. Available at arXiv:Astro-ph/0501343.
  • Friedman, J., Hastie, T. and Tibshirani, R. (2010). Regularization paths for generalized linear models via coordinate descent. J. Stat. Softw. 33 1–22.
  • Green, P. J. (1995). Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika 82 711–732.
  • Grünwald, P. D., Myung, I. J. and Pitt, M. A. (2005). Advances in Minimum Description Length: Theory and Applications. MIT press, Cambridge.
  • Güdel, M. (2004). X-ray astronomy of stellar coronae. Astron. Astrophys. Rev. 12 71–237.
  • Kashyap, V. L., Drake, J. J., Güdel, M. and Audard, M. (2002). Flare heating in stellar coronae. Astrophys. J. 580 1118–1132.
  • Korhonen, H., Berdyugina, S. V., Hackman, T., Duemmler, R., Ilyin, I. V. and Tuominen, I. (1999). Study of FK Comae Berenices. I. Surface images for 1994 and 1995. Astron. Astrophys. 346 101–110.
  • Lai, T. L. and Xing, H. (2011). A simple Bayesian approach to multiple change-points. Statist. Sinica 21 539–569.
  • Leclerc, Y. G. (1989). Constructing simple stable descriptions for image partitioning. Int. J. Comput. Vis. 3 73–102.
  • Lee, T. C. M. (1997). Some models and methods in image segmentation. Ph.D. thesis, Macquarie Univ., Sydney, Australia.
  • Lee, T. C. M. (1998). Segmenting images corrupted by correlated noise. IEEE Trans. Pattern Anal. Mach. Intell. 20 481–492.
  • Lee, T. C. M. (2002a). Automatic smoothing for discontinuous regression functions. Statist. Sinica 12 823–842.
  • Lee, T. C. M. (2002b). On algorithms for ordinary least squares regression spline fitting: A comparative study. J. Stat. Comput. Simul. 72 647–663.
  • Lee, H., Kashyap, V. L., van Dyk, D. A., Connors, A., Drake, J. J., Izem, R., Meng, X. L., Min, S., Park, T., Ratzlaff, P., Siemiginowska, A. and Zezas, A. (2011). Accounting for calibration uncertainties in X-ray analysis: Effective areas in spectral fitting. Astrophys. J. 731 126–144.
  • Loader, C. R. (1992). A log-linear model for a Poisson process change point. Ann. Statist. 20 1391–1411.
  • MacKay, D. J. C. (2003). Information Theory, Inference, and Algorithms. Cambridge Univ. Press, Cambridge, UK.
  • McKee, M. (2012). Superflares’ erupt on some Sun-like stars. Nature News, May 16, 2012.
  • Mei, Y., Han, S. W. and Tsui, K.-L. (2011). Early detection of a change in Poisson rate after accounting for population size effects. Statist. Sinica 21 597–624.
  • Moreno, E., Casella, G. and Garcia-Ferrer, A. (2005). An objective Bayesian analysis of the change point problem. Stoch. Environ. Res. Risk Assess. 19 191–204.
  • Ninomiya, Y. (2015). Change-point model selection via AIC. Ann. Inst. Statist. Math. 67 943–961.
  • Park, J. H. (2010). Structural change in US presidents’ use of force. Amer. J. Polit. Sci. 54 766–782.
  • Park, T., Krafty, R. T. and Sánchez, A. I. (2012). Bayesian semi-parametric analysis of Poisson change-point regression models: Application to policy-making in Cali, Colombia. J. Appl. Stat. 39 2285–2298.
  • Park, T., Kashyap, V., Siemiginowska, A., van Dyk, D. A., Zezas, A., Heinke, C. and Wargelin, B. J. (2006). Bayesian estimation of hardness ratios: Modeling and computations. Astrophys. J. 652 610–628.
  • Raftery, A. E. and Akman, V. E. (1986). Bayesian analysis of a Poisson process with a change-point. Biometrika 73 85–89.
  • Rissanen, J. (1989). Stochastic Complexity in Statistical Inquiry. World Scientific, Teaneck, NJ.
  • Rissanen, J. (2007). Information and Complexity in Statistical Modeling. Springer, New York.
  • Ruppert, D., Wand, M. P. and Carroll, R. J. (2003). Semiparametric Regression. Cambridge Univ. Press, Cambridge.
  • Scargle, J. D. (1998). Studies in astronomical time series analysis. V. Bayesian blocks, a new method to analyze structure in photon counting data. Astrophys. J. 504 405–418.
  • Scargle, J. D., Norris, J. P., Jackson, B. and Chiang, J. (2013). Studies in astronomical time series analysis. VI. Bayesian block representations. Astrophys. J. 764 167–192.
  • Shen, J. J. and Zhang, N. R. (2012). Change-point model on nonhomogeneous Poisson processes with application in copy number profiling by next-generation DNA sequencing. Ann. Appl. Stat. 6 476–496.
  • Strassmeier, K. G. (2009). Starspots. Astron. Astrophys. Rev. 17 251–308.
  • Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. J. Roy. Statist. Soc. Ser. B 58 267–288.
  • van Dyk, D. A. and Kang, H. (2004). Highly structured models for spectral analysis in high-energy astrophysics. Statist. Sci. 19 275–293.
  • van Dyk, D. A., Connors, A., Kashyap, V. and Siemiginowska, A. (2001). Analysis of energy spectra with low photon counts via Bayesian posterior simulation. Astrophys. J. 548 224–243.
  • van Dyk, D., Connors, A., Esch, D. N., Freeman, P., Kang, H., Karovska, M., Kashyap, V., Siemiginowska, A. and Zezas, A. (2006). Deconvolution in high-energy astrophysics: Science, instrumentation, and methods. Bayesian Anal. 1 189–235.
  • Worsley, K. J. (1986). Confidence regions and test for a change-point in a sequence of exponential family random variables. Biometrika 73 91–104.
  • Xu, J., van Dyk, D. A., Kashyap, V. L., Siemiginowska, A., Connors, A., Drake, J. J., Meng, X. L., Ratzlaff, P. and Yu, Y. (2014). A fully Bayesian method for jointly fitting instrumental calibration and X-ray spectral models. Astrophys. J. 794 97 (21 pages).
  • Yao, Y.-C. (1988). Estimating the number of change-points via Schwarz’ criterion. Statist. Probab. Lett. 6 181–189.
  • Yau, C. Y., Tang, C. M. and Lee, T. C. M. (2015). Estimation of multiple-regime threshold autoregressive models with structural breaks. J. Amer. Statist. Assoc. 110 1175–1186.
  • Zhang, N. R. and Siegmund, D. O. (2007). A modified Bayes information criterion with applications to the analysis of comparative genomic hybridization data. Biometrics 63 22–32.