Adaptive tests for periodic signal detection with applications to laser vibrometry

Magalie Fromont; Céline Lévy-leduc

ESAIM: Probability and Statistics (2006)

  • Volume: 10, page 46-75
  • ISSN: 1292-8100

Abstract

top
Initially motivated by a practical issue in target detection via laser vibrometry, we are interested in the problem of periodic signal detection in a Gaussian fixed design regression framework. Assuming that the signal belongs to some periodic Sobolev ball and that the variance of the noise is known, we first consider the problem from a minimax point of view: we evaluate the so-called minimax separation rate which corresponds to the minimal l2-distance between the signal and zero so that the detection is possible with prescribed probabilities of error. Then, we propose a testing procedure which is available when the variance of the noise is unknown and which does not use any prior information about the smoothness degree or the period of the signal. We prove that it is adaptive in the sense that it achieves, up to a possible logarithmic factor, the minimax separation rate over various periodic Sobolev balls simultaneously. The originality of our approach as compared to related works on the topic of signal detection is that our testing procedure is sensitive to the periodicity assumption on the signal. A simulation study is performed in order to evaluate the effect of this prior assumption on the power of the test. We do observe the gains that we could expect from the theory. At last, we turn to the application to target detection by laser vibrometry that we had in view.

How to cite

top

Fromont, Magalie, and Lévy-leduc, Céline. "Adaptive tests for periodic signal detection with applications to laser vibrometry." ESAIM: Probability and Statistics 10 (2006): 46-75. <http://eudml.org/doc/249739>.

@article{Fromont2006,
abstract = { Initially motivated by a practical issue in target detection via laser vibrometry, we are interested in the problem of periodic signal detection in a Gaussian fixed design regression framework. Assuming that the signal belongs to some periodic Sobolev ball and that the variance of the noise is known, we first consider the problem from a minimax point of view: we evaluate the so-called minimax separation rate which corresponds to the minimal l2-distance between the signal and zero so that the detection is possible with prescribed probabilities of error. Then, we propose a testing procedure which is available when the variance of the noise is unknown and which does not use any prior information about the smoothness degree or the period of the signal. We prove that it is adaptive in the sense that it achieves, up to a possible logarithmic factor, the minimax separation rate over various periodic Sobolev balls simultaneously. The originality of our approach as compared to related works on the topic of signal detection is that our testing procedure is sensitive to the periodicity assumption on the signal. A simulation study is performed in order to evaluate the effect of this prior assumption on the power of the test. We do observe the gains that we could expect from the theory. At last, we turn to the application to target detection by laser vibrometry that we had in view. },
author = {Fromont, Magalie, Lévy-leduc, Céline},
journal = {ESAIM: Probability and Statistics},
keywords = { Periodic signal detection; adaptive test; minimax separation rates; nonparametric regression.; periodic signal detection; minimax separation rates},
language = {eng},
month = {1},
pages = {46-75},
publisher = {EDP Sciences},
title = {Adaptive tests for periodic signal detection with applications to laser vibrometry},
url = {http://eudml.org/doc/249739},
volume = {10},
year = {2006},
}

TY - JOUR
AU - Fromont, Magalie
AU - Lévy-leduc, Céline
TI - Adaptive tests for periodic signal detection with applications to laser vibrometry
JO - ESAIM: Probability and Statistics
DA - 2006/1//
PB - EDP Sciences
VL - 10
SP - 46
EP - 75
AB - Initially motivated by a practical issue in target detection via laser vibrometry, we are interested in the problem of periodic signal detection in a Gaussian fixed design regression framework. Assuming that the signal belongs to some periodic Sobolev ball and that the variance of the noise is known, we first consider the problem from a minimax point of view: we evaluate the so-called minimax separation rate which corresponds to the minimal l2-distance between the signal and zero so that the detection is possible with prescribed probabilities of error. Then, we propose a testing procedure which is available when the variance of the noise is unknown and which does not use any prior information about the smoothness degree or the period of the signal. We prove that it is adaptive in the sense that it achieves, up to a possible logarithmic factor, the minimax separation rate over various periodic Sobolev balls simultaneously. The originality of our approach as compared to related works on the topic of signal detection is that our testing procedure is sensitive to the periodicity assumption on the signal. A simulation study is performed in order to evaluate the effect of this prior assumption on the power of the test. We do observe the gains that we could expect from the theory. At last, we turn to the application to target detection by laser vibrometry that we had in view.
LA - eng
KW - Periodic signal detection; adaptive test; minimax separation rates; nonparametric regression.; periodic signal detection; minimax separation rates
UR - http://eudml.org/doc/249739
ER -

References

top
  1. Y. Baraud, Non-asymptotic minimax rates of testing in signal detection. Bernoulli8 (2002) 577–606.  Zbl1007.62042
  2. Y. Baraud, S. Huet, and B. Laurent, Adaptive tests of linear hypotheses by model selection. Ann. Statist.31 (2003) 225–251.  Zbl1018.62037
  3. L. Birgé, An alternative point of view on Lepski's method, in State of the Art in Probability and Statistics (Leiden, 1999), 113–133, IMS Lecture Notes Monogr. Ser.36 (2000).  
  4. P.J. Brockwell and R.A. Davis, Time series: theory and methods. Springer Series in Statistics. Springer-Verlag, New York, second edition (1991).  Zbl0709.62080
  5. R. Eubank and J. Hart, Testing goodness-of-fit in regression via order selection criteria. Ann. Stat.20 (1992) 1412–1425.  Zbl0776.62045
  6. J. Fan and Q. Yao, Nonlinear Time series. Springer series in Statistics. Springer-Verlag, New York, Nonparametric and parametric methods (2003).  
  7. G. Gayraud and C. Pouet, Minimax testing composite null hypotheses in the discrete regression scheme. Math. Methods Stat.10 (2001) 375–394.  Zbl1005.62048
  8. P. Gregory and T. Loredo, A new method for the detection of a periodic signal of unknown shape and period. The Astrophysical J.398 (1992) 146–168.  Zbl0925.62026
  9. W. Härdle and A. Kneip, Testing a regression model when we have smooth alternatives in mind. Scand. J. Stat.26 (1999) 221–238.  Zbl0934.62043
  10. J. Horowitz and V. Spokoiny, An adaptive, rate-optimal test of a parametric mean-regression model against a nonparametric alternative. Econometrica69 (2001) 599–631.  Zbl1017.62012
  11. Y. Ingster, Minimax nonparametric detection of signals in white Gaussian noise. Probl. Inf. Transm.18 (1982) 130–140.  Zbl0499.94002
  12. Y. Ingster, Asymptotically minimax testing for nonparametric alternatives I-II-III. Math. Methods Statist.2 (1993) 85–114, 171–189, 249–268.  Zbl0798.62057
  13. B. Laurent and P. Massart, Adaptive estimation of a quadratic functional by model selection. Ann. Statist.28 (2000) 1302–1338.  Zbl1105.62328
  14. M. Lavielle and C. Lévy-Leduc, Semiparametric estimation of the frequency of unknown periodic functions and its application to laser vibrometry signals. IEEE Trans. Signal Proces.53 (2005) 2306–2314.  
  15. O. Lepski and V. Spokoiny, Minimax nonparametric hypothesis testing: The case of an inhomogeneous alternative. Bernoulli5 (1999) 333–358.  Zbl0946.62050
  16. O. Lepski and A. Tsybakov, Asymptotically exact nonparametric hypothesis testing in sup-norm and at a fixed point. Probab. Theory Relat. Fields117 (2000) 17–48.  Zbl0971.62022
  17. M. Prenat, Vibration modes and laser vibrometry performance in noise, in Proceedings of the Physics in Signal and Image Processing conference (PSIP'01), 23–24 janvier 2001, Marseille, France (2001).  
  18. B.G. Quinn and E.J. Hannan, The estimation and tracking of frequency. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge (2001).  Zbl0969.62060
  19. V. Spokoiny, Adaptive hypothesis testing using wavelets. Ann. Stat.24 (1996) 2477–2498.  Zbl0898.62056
  20. V. Spokoiny, Adaptive and spatially adaptive testing of a nonparametric hypothesis. Math. Methods Stat.7 (1998) 245–273.  Zbl1103.62345

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.