A Monte Carlo subsampling method for estimating the distribution of signal-to-noise ratio statistics in nonparametric ti
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A Monte Carlo subsampling method for estimating the distribution of signal-to-noise ratio statistics in nonparametric time series regression models Francesco Giordano1 · Pietro Coretto1 Accepted: 19 August 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract Signal-to-noise ratio (SNR) statistics play a central role in many applications. A common situation where SNR is studied is when a continuous time signal is sampled at a fixed frequency with some noise in the background. While estimation methods exist, little is known about its distribution when the noise is not weakly stationary. In this paper we develop a nonparametric method to estimate the distribution of an SNR statistic when the noise belongs to a fairly general class of stochastic processes that encompasses both short and long-range dependence, as well as nonlinearities. The method is based on a combination of smoothing and subsampling techniques. Computations are only operated at the subsample level, and this allows to manage the typical enormous sample size produced by modern data acquisition technologies. We derive asymptotic guarantees for the proposed method, and we show the finite sample performance based on numerical experiments. Finally, we propose an application to electroencephalography data. Keywords Random subsampling · Nonparametric smoothing · Kernel regression · Time series data · Stochastic processes
1 Introduction Signal-to-noise ratio (SNR) statistics are widely used to describe the strength of the variations of the signal relative to those expressed by the noise. SNR statistics are used to quantify diverse aspects of models where an observable quantity Y is decom-
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Pietro Coretto [email protected] Francesco Giordano [email protected]
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Department of Economics and Statistics, University of Salerno, Via Giovanni Paolo II, no. 132, 84084 Fisciano, SA, Italy
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F. Giordano, P. Coretto
posed into a predictable or structural component s, often called signal or model, and a stochastic component ε, called noise or error. Although the definition of SNR is rather general in this paper we focus on a typical situation where one assumes a sequence {Yi }i∈Z is determined by Yi := s(ti ) + εi ,
(1)
where i is a time index, s(·) is a smooth function of time evaluated at the time point ti with i ∈ Z, and {εi }i∈Z is some random sequence. Assume ti ∈ (0, 1), however if a time series is observed at time points ti ∈ (a, b), these can be rescaled onto the interval (0, 1) without changing the results of this paper. Equation (1) is a popular model in many applications that range from physical sciences to engineering, biosciences, social sciences, etc. (see Parzen 1966, 1999 and references therein). Although we use the conventional term “noise” for εi , this term may have a rich structure well beyond what we would usually consider noise. Some of the terminology here originates from physical sciences where the following concepts have been first explored. Consider a non stochastic signal s(t) defined on the time 1 interval (
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