SURE shrinkage of Gaussian paths and signal identification

SURE shrinkage of Gaussian paths and signal identification Nicolas Privault∗ Department of Mathematics City University of Hong Kong Tat Chee Avenue Kowloon Tong Hong Kong Anthony R´eveillac† Institut f¨ur Mathematik Humboldt-Universit¨at zu Berlin Unter den Linden 6 10099 Berlin Germany October 29, 2018 Abstract Using integration by parts on Gaussian space we construct a Stein Unbiased Risk Estimator (SURE) for the drift of Gaussian processes using their local and occupation times. By almost-sure minimization of the SURE risk of shrinkage estimators we derive an estimation and de-noising procedure for an input signal perturbed by a continuous-time Gaussian noise. Key words: Estimation, SURE shrinkage, thresholding, denoising, Gaussian processes, Malliavin calculus. Mathematics Subject Classification: 93E10, 93E14, 60G35, 60H07. 1 Introduction Let X be a Gaussian random vector on Rd with unknown mean m and known covari- ance matrix σ2Id under a probability measure Pm. It is well-known [13] that given g : Rd →Rd a sufficiently smooth function, the mean square risk ∥X + g(X) −m∥2 Rd of X + g(X) to m can be estimated unbiasedly by SURE := σ2d + d X i=1 gi(X)2 + 2 d X i=1 ∇ig(X), (1.1) ∗nprivaul@cityu.edu.hk †anthony.reveillac@univ-lr.fr 1 arXiv:0809.1516v2 [math.ST] 23 Feb 2009 from the identity IEm  ∥X + g(X) −m∥2 Rd  = σ2d + IEm " d X i=1 gi(X)2 + 2 d X i=1 ∇ig(X) # (1.2) which is obtained by Gaussian integration by parts under Pm. The estimator (1.1), which is independent of m, is called the Stein Unbiased Risk Estimate (SURE). When (gλ)λ∈Λ is a family of functions it makes sense to almost surely minimize the Stein Unbiased Risk Estimate (1.1) of gλ with respect to the parameter λ. This point of view has been developed by Donoho and Johnstone [4] for the design of spatially adaptive estimators by shrinkage of wavelet coefficients of noisy data via X + gλ(X) = λη(X/λ), where η(x) is a threshold function. In this paper we construct a Stein type Unbiased Risk Estimator for the deterministic drift (ut)t∈R+ of a one dimensional Gaussian processes (Xt)t∈[0,T] via an extension of the identity (1.2) introduced in [10], [9] on the Wiener space. For example, given α(t) and λ(t) two functions given in parametric form, the SURE risk of the estimator Xt + ξα,λ t (Xt) = α(t) + λ(t)ηS Xt −α(t) λ(t)  , t ∈[0, T], where ηH is the hard threshold function (5.1) below, is given by SURE (X + ξα,λ(X)) = T + Z T 0 (Xt −α(t))2 γ(t, t) 1{|Xt−α(t)|≤λ√ γ(t,t)}dt + 2λ¯ℓλ T −2¯Lλ T, where γ(s, t) = Cov(Xs, Xt), 0 ≤s, t ≤T, denotes the covariance of (Xt)t∈[0,T] and ¯ℓλ T, ¯Lλ T respectively denote the local and occupation time of (|Xt −α(t)|/ p γ(t, t))t∈[0,T], cf. Proposition 5.1. We apply this technique to de-noising and identification of the input signal in a Gaussian channel via the minimization of SURE (X +ξα,λ(X)). This 2 yields in particular an estimator of the drift of Xt from the estimation of α(t), and an optimal noise removal threshold from the estimation of λ. This approach differs from classical signal detection techniques which usually rely on likelihood ratio tests, cf e.g. [8], Chapter VI. It also requires an a priori hypothesis on the parametric form of α(t). We proceed as follows. In Section 2 we recall our framework of functional estimation of drift trajectories. In Section 3 we derive Stein’s unbiased risk estimate for the estimation of the drift of Gaussian processes. In Section 4 we discuss its application to soft thresholding for Gaussian processes using the local time and obtain an upper bound for the risk of such estimators. We also show the existence of an optimal parameter and the smoothness of the risk function. In Section 5 we consider the case of hard thresholding. In Section 6 we consider several numerical examples where α(t) is given in parametric form. In Section 7 we recall some elements of stochastic analysis of Gaussian processes. 2 Functional drift estimation In this section we recall the setting of functional drift estimation to be used in this pa- per. Given T > 0 we consider a real-valued centered Gaussian process X = (Xt)t∈[0,T] with non-vanishing covariance function γ(s, t) = IE[XsXt], s, t ∈[0, T], on a probability space (Ω, F, P), where (F)t∈[0,T] is the filtration generated by (Xt)t∈[0,T]. Assume that under a probability measure Pu we observe the paths of (Xt)t∈[0,T] de- composed as Xt = ut + Xu t , t ∈[0, T], where u = (ut)t∈[0,T] is a square integrable F-adapted process and (Xu t )t∈[0,T] is a centered Gaussian process with covariance γ(s, t) = IEu[Xu s , Xu t ], 0 ≤s, t ≤T, 3 where IEu denotes the expectation under Pu. Given a continuous time observation of the process (Xt)t∈[0,T] we will propose estimators of the unknown drift function u. Definition 2.1. The risk of an estimator ξ := (ξt)t∈[0,T] to u is defined as R(γ, µ, ξ) := IEu Z T 0 |ξt −ut|2µ(dt)  where µ is a positive measure on [0, T]. Examples of risk measures µ include the Lebesgue measure and µ(dt) = n X i=1 aiδti(dt), a1, . . . ,

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