Self-Similarity: Part II—Optimal Estimation of Fractal Processes

In a companion paper (see Self-Similarity: Part I-Splines and Operators), we characterized the class of scale-invariant convolution operators: the generalized fractional derivatives of order gamma. We used these operators to specify regularization functionals for a series of Tikhonov-like least-squares data fitting problems and proved that the general solution is a fractional spline of twice the order. We investigated the deterministic properties of these smoothing splines and proposed a fast Fourier transform (FFT)-based implementation. Here, we present an alternative stochastic formulation to further justify these fractional spline estimators. As suggested by the title, the relevant processes are those that are statistically self-similar; that is, fractional Brownian motion (fBm) and its higher order extensions. To overcome the technical difficulties due to the nonstationary character of fBm, we adopt a distributional formulation due to Gel'fand. This allows us to rigorously specify an innovation model for these fractal processes, which rests on the property that they can be whitened by suitable fractional differentiation. Using the characteristic form of the fBm, we then derive the conditional probability density function (PDF) p(B_H(t)|Y), where Y={B_H(k)+n[k]}_kisinZ are the noisy samples of the fBm B_H(t) with Hurst exponent H. We find that the conditional mean is a fractional spline of degree 2H, which proves that this class of functions is indeed optimal for the estimation of fractal-like processes. The result also yields the optimal [minimum mean-square error (MMSE)] parameters for the smoothing spline estimator, as well as the connection with kriging and Wiener filtering

I. Introduction

In the preceding paper [1], we demonstrated the power of the differential formulation of splines by constructing an extended family of fractional splines. These functions are specified in terms of a differential operator L, which, in the present case, is constrained to be scale invariant (or self-similar). We also investigated an alternative variational formulation which allowed us to recover a subset of these splines (the ones associated with self-adjoint operators) based on the minimization of some scale-invariant “spline energy” involving the same type of operator. We used this deterministic framework to specify a general parametric class of smoothing spline estimators for fitting discrete signals corrupted by noise.