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21
Nonparametric estimation of a periodic function
 Biometrika
, 2000
"... ABSTRACT. Motivated by applications to brightness data on periodic variable stars, we study nonparametric methods for estimating both the period and the amplitude function from noisy observations of a periodic function made at irregularly spaced times. It is shown that nonparametric estimators of pe ..."
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Cited by 14 (1 self)
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ABSTRACT. Motivated by applications to brightness data on periodic variable stars, we study nonparametric methods for estimating both the period and the amplitude function from noisy observations of a periodic function made at irregularly spaced times. It is shown that nonparametric estimators of period converge at parametric rates and attain a semiparametric lower bound which is the same if the shape of the periodic function is unknown as if it were known. Also, firstorder properties of nonparametric estimators of the amplitude function are identical to those that would obtain if the period were known. Numerical simulations and applications to real data show the method to work well in practice. KEY WORDS AND PHRASES. frequency estimation, nonparametric regression, semiparametric estimation, NadarayaWatson estimator, MACHO project, variable star data. SHORT TITLE. Estimation of a periodic function
DataAdaptive Wavelets and MultiScale SingularSpectrum Analysis
, 2000
"... Using multiscale ideas from wavelet analysis, we extend singularspectrum analysis (SSA) to the study of nonstationary time series, including the case where intermittency gives rise to the divergence of their variance. The wavelet transform resembles a local Fourier transform within a finite moving ..."
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Cited by 8 (1 self)
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Using multiscale ideas from wavelet analysis, we extend singularspectrum analysis (SSA) to the study of nonstationary time series, including the case where intermittency gives rise to the divergence of their variance. The wavelet transform resembles a local Fourier transform within a finite moving window whose width W , proportional to the major period of interest, is varied to explore a broad range of such periods. SSA, on the other hand, relies on the construction of the lagcorrelation matrix C on M lagged copies of the time series over a fixed window width W to detect the regular part of the variability in that window in terms of the minimal number of oscillatory components; here W = M#t with #t as the time step. The proposed multiscale SSA is a local SSA analysis within a movingwindowof width M # W # N , whereN is the length of the time series. Multiscale SSA varies W , while keeping a fixed W/M ratio, and uses the eigenvectors of the corresponding lagcorrelation matrix...
Statistics and Music: Fitting a Local Harmonic Model to Musical Sound Signals
, 1998
"... Statistical modeling and analysis have been applied to different music related fields. One of them is sound synthesis and analysis. Sound can be represented as a realvalued function of time. This function can be sampled at a small enough rate so that the resulting discrete version is almost as goo ..."
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Cited by 8 (4 self)
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Statistical modeling and analysis have been applied to different music related fields. One of them is sound synthesis and analysis. Sound can be represented as a realvalued function of time. This function can be sampled at a small enough rate so that the resulting discrete version is almost as good as the continuous one. This permits one to study musical sounds as a discrete time series, an entity for whichmany statistical techniques are available. Physical modeling suggests that manymusical instruments' sounds are characterized bya harmonic and an additive noise signal. The noise is not something to get rid of rather it's an important part of the signal. In this research the interest is in separating these two elements of the sound. To do so a local harmonic model that tracks ch...
Advanced Spectral Analysis Methods
 Provenzale (Eds.), Past and Present Variability of the SolarTerrestrial System: Measurement, Data Analysis and Theoretical Models, Societá Italiana di Fisica/IOS Press, Bologna/Amsterdam
, 1997
"... Introduction The purpose of timeseries analysis is to detect basic properties of the system that engenders a time series. The hope of predicting the system's future evolution is closely related to the possibility of such detection. The most easily predictable components of a system's evolution are ..."
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Cited by 6 (2 self)
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Introduction The purpose of timeseries analysis is to detect basic properties of the system that engenders a time series. The hope of predicting the system's future evolution is closely related to the possibility of such detection. The most easily predictable components of a system's evolution are the regular, deterministic ones; hence we look for trends and periodic oscillations. In doing so, it is often convenient to move from the time domain to the frequency domain. The frequency domain approach to timeseries analysis is based on the Wiener [1] Khinchin [2] theorem, which states the equality between the power spectrum and the Fourier transform of the autocorrelation function (ACF) of a time series. Many techniques for spectral analysis of discretely sampled time series have been developed. In this chapter we aim to: a) review a few commonly used spectral estimation techniques (see Table I in Sec. 7); and b) highlight the strengths and weaknesses of these techniques. The natural
Local Harmonic Estimation in Musical Sound Signals
 Journal of the American Statistical Association
, 2001
"... this paper the interest is in separating these two elements of the sound and finding parametric representations with musical meaning. To do so a local harmonic model that tracks changes in pitch and in the amplitudes of the harmonics is fit. Deterministic changes in the signal, such as pitch change, ..."
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Cited by 6 (0 self)
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this paper the interest is in separating these two elements of the sound and finding parametric representations with musical meaning. To do so a local harmonic model that tracks changes in pitch and in the amplitudes of the harmonics is fit. Deterministic changes in the signal, such as pitch change, suggest that different temporal window sizes should be considered. Ways to choose appropriate window sizes are studied. Amongst other things our analysis provides estimates of the harmonic signal and of the noise signal. Different musical composition applications may be based on the estimates. KEY WORDS: Musical Sound Signals, Local Harmonic Model, Widow Size Selection 1. INTRODUCTION Statistics has been applied in various ways to music. For example, various stochastic techniques have been applied in composition (Jones 1981). Stochastic techniques have also been used in forecasting unfinished works (Dirst and Weigend 1992). Voss and Clarke (1975) studied the spectral properties of different musical signals and speculated on the possibility of it being so called 1/f noise. In Brillinger and Irizarry (1998) this is studied in more detail, and in particular higher order statistics are examined. In this paper the particular application that will be examined in detail is the analysis of sound signals produced by musical instruments. In this field, statistical techniques have been used, for example, to separate the signals into what are believed to be deterministic and stochastic parts and to deconstruct the deterministic part into harmonic components.
A Fourier analysis of extreme events
 SUBMITTED TO THE BERNOULLI
, 2013
"... The extremogram is an asymptotic correlogram for extreme events constructed from a regularly varying stationary sequence. In this paper, we define a frequency domain analog of the correlogram: a periodogram generated from a suitable sequence of indicator functions of rare events. We derive basic pro ..."
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Cited by 2 (2 self)
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The extremogram is an asymptotic correlogram for extreme events constructed from a regularly varying stationary sequence. In this paper, we define a frequency domain analog of the correlogram: a periodogram generated from a suitable sequence of indicator functions of rare events. We derive basic properties of the periodogram such as the asymptotic independence at the Fourier frequencies and use this property to show that weighted versions of the periodogram are consistent estimators of a spectral density derived from the extremogram.
2000: Dataadaptive wavelets and multiscale SSA
 Physica D
"... Using multiscale ideas from wavelet analysis, we extend singularspectrum analysis (SSA) to the study of nonstationary time series of length N whose intermittency can give rise to the divergence of their variance. The wavelet transform is a kind of local Fourier transform within a finite moving win ..."
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Cited by 1 (1 self)
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Using multiscale ideas from wavelet analysis, we extend singularspectrum analysis (SSA) to the study of nonstationary time series of length N whose intermittency can give rise to the divergence of their variance. The wavelet transform is a kind of local Fourier transform within a finite moving window whose width W, proportional to the major period of interest, is varied to explore a broad range of such periods. SSA, on the other hand, relies on the construction of the lagcovariance matrix C on M lagged copies of the time series over a fixed window width W to detect the regular part of the variability in that window in terms of the minimal number of oscillatory components; here W = M∆t, with ∆t the time step. The proposed multiscale SSA is a local SSA analysis within a moving window of width M ≤ W ≤ N. Multiscale SSA varies W, while keeping a fixed W/M ratio, and uses the eigenvectors of the corresponding lagcovariance matrix CM as a dataadaptive wavelets; successive eigenvectors of CM correspond approximately to successive derivatives of the first mother wavelet in standard wavelet analysis. Multiscale SSA thus solves objectively the delicate problem of optimizing the analyzing wavelet in the timefrequency domain, by a suitable localization of the signal’s covariance matrix. We present several examples of application to synthetic signals with fractal or powerlaw behavior
Asymptotic Distribution Of Estimates For A TimeVarying Parameter In A Harmonic Model With Multiple Fundamentals
 Statistica Sinica
, 2001
"... : Windowbased estimates for stochastic harmonic regression models are useful for cases where harmonic parameters appear to be timevarying. Least squares estimates for harmonic models with one fundamental have been studied and asymptotic variance expressions have been developed. This paper extends ..."
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: Windowbased estimates for stochastic harmonic regression models are useful for cases where harmonic parameters appear to be timevarying. Least squares estimates for harmonic models with one fundamental have been studied and asymptotic variance expressions have been developed. This paper extends these results to weighted least squares for the multiple fundamental case, and presents an application in signal processing. Key words and phrases: Asymptotic variance, harmonic regression, signal processing, sound analysis, timevarying parameters, weighted least squares estimates. 1. Introduction Consider the signal plus noise model y t = s(t; fi) + ffl t ; (t = 1; : : : ; T ); (1) where the signal s(t; fi) is composed of J periodic components s(t; fi) = J X j=1 s j (t; fi j ); fi = (fi 1 ; : : : ; fi J ) 0 (2) and each component s j (t; fi j ) is a sum of K j sinusoidal components s j (t; fi j ) = K j X k=1 fA j;k cos(! j;k t) +B j;k sin(! j;k t)g (3) fi j = (A j;1 ; B j;1 ;...