Bernoulli

• Bernoulli
• Volume 21, Number 2 (2015), 1134-1165.

On detecting harmonic oscillations

Abstract

In this paper, we focus on the following testing problem: assume that we are given observations of a real-valued signal along the grid $0,1,\ldots,N-1$, corrupted by white Gaussian noise. We want to distinguish between two hypotheses: (a) the signal is a nuisance – a linear combination of $d_{n}$ harmonic oscillations of known frequencies, and (b) signal is the sum of a nuisance and a linear combination of a given number $d_{s}$ of harmonic oscillations with unknown frequencies, and such that the distance (measured in the uniform norm on the grid) between the signal and the set of nuisances is at least $\rho>0$. We propose a computationally efficient test for distinguishing between (a) and (b) and show that its “resolution” (the smallest value of $\rho$ for which (a) and (b) are distinguished with a given confidence $1-\alpha$) is $\mathrm{O}(\sqrt{\ln(N/\alpha)/N})$, with the hidden factor depending solely on $d_{n}$ and $d_{s}$ and independent of the frequencies in question. We show that this resolution, up to a factor which is polynomial in $d_{n}$, $d_{s}$ and logarithmic in $N$, is the best possible under circumstances. We further extend the outlined results to the case of nuisances and signals close to linear combinations of harmonic oscillations, and provide illustrative numerical results.

Article information

Source
Bernoulli, Volume 21, Number 2 (2015), 1134-1165.

Dates
First available in Project Euclid: 21 April 2015

https://projecteuclid.org/euclid.bj/1429624973

Digital Object Identifier
doi:10.3150/14-BEJ600

Mathematical Reviews number (MathSciNet)
MR3338659

Zentralblatt MATH identifier
06445970

Citation

Juditsky, Anatoli; Nemirovski, Arkadi. On detecting harmonic oscillations. Bernoulli 21 (2015), no. 2, 1134--1165. doi:10.3150/14-BEJ600. https://projecteuclid.org/euclid.bj/1429624973

References

• [1] Chiu, S.-T. (1989). Detecting periodic components in a white Gaussian time series. J. Roy. Statist. Soc. Ser. B 51 249–259.
• [2] Davies, R.B. (1987). Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika 74 33–43.
• [3] Djuric, P.M. (1996). A model selection rule for sinusoids in white Gaussian noise. IEEE Trans. Signal Process. 44 1744–1751.
• [4] Fisher, R.A. (1929). Test of significance in harmonic analysis. Proc. R. Soc. Lond. Ser. A 125 54–59.
• [5] Goldenshluger, A. and Nemirovski, A. (1997). Adaptive de-noising of signals satisfying differential inequalities. IEEE Trans. Inform. Theory 43 872–889.
• [6] Hannan, E.J. (1970). Multiple Time Series. New York: Wiley.
• [7] Hannan, E.J. (1993). Determining the number of jumps in a spectrum. In Developments in Time Series Analysis (T.S. Rao, ed.) 127–138. London: Chapman & Hall.
• [8] Juditsky, A. and Nemirovski, A. (2009). Nonparametric denoising of signals with unknown local structure, I: Oracle inequalities. Appl. Comput. Harmon. Anal. 27 157–179.
• [9] Juditsky, A. and Nemirovski, A. (2010). Nonparametric denoising signals of unknown local structure, II: Nonparametric function recovery. Appl. Comput. Harmon. Anal. 29 354–367.
• [10] Kavalieris, L. and Hannan, E.J. (1994). Determining the number of terms in a trigonometric regression. J. Time Series Anal. 15 613–625.
• [11] Nadler, B. and Kontorovich, A. (2011). Model selection for sinusoids in noise: Statistical analysis and a new penalty term. IEEE Trans. Signal Process. 59 1333–1345.
• [12] Nemirovskiĭ, A.S. (1981). Prediction under conditions of indeterminacy. Problems Inform. Transmission 17 73–83 (in Russian).
• [13] Nemirovskiĭ, A.S. (1992). On nonparametric estimation of functions satisfying differential inequalities. In Topics in Nonparametric Estimation (R. Khasminskii, ed.). Adv. Soviet Math. 12 7–43. Providence, RI: Amer. Math. Soc.
• [14] Pisarenko, V.F. (1973). The retrieval of harmonics from a covariance function. Geophys. J. Roy. Astron. Soc. 33 347–366.
• [15] Quinn, B.G. and Hannan, E.J. (2001). The Estimation and Tracking of Frequency. Cambridge Series in Statistical and Probabilistic Mathematics 9. Cambridge: Cambridge Univ. Press.
• [16] Quinn, B.G. and Kotsookos, P.J. (1994). Threshold behavior of the maximum likelihood estimator of frequency. IEEE Trans. Signal Process. 42 3291–3294.
• [17] Rife, D.C. and Boorstyne, R.R. (1974). Single-tone parameter estimation from discrete-time observations. IEEE Trans. Inform. Theory 20 591–598.
• [18] Schmidt, R.O. (1986). Multiple emitter location and signal parameter estimation. IEEE Trans. Antennas and Propagation 34 276–280.
• [19] Stoica, P. and Moses, R.L. (1997). Introduction to Spectral Analysis. New York: Prentice-Hall.
• [20] Whittle, P. (1954). The statistical analysis of a seiche record. J. Marine Res. 13 76–100.