Fractional Splines and Fractals

A MATLAB package is made available for computing the fractional smoothing spline estimator of a 1D signal, and for generating fBms (fractional Brownian motion).

Fractional Brownian motion processes, which were introduced by Mandelbrot in 1968, are non-stationary stochastic processes that are statistically self-similar (fractals). We have shown recently that fractional splines and fBms are intimately connected: they can be defined as solution of the same type of (fractional) differential equation. In the first case, the excitation is a deterministic stream of Dirac impulses (discrete signal), while in the second, it is white Gaussian noise. Thanks to this connection, we could prove that a smoothing spline estimator with an appropriate set of parameters (spline order or degree and regularization factor) produces the minimum mean squares error reconstruction of a fBm signal corrupted by additive noise.

The package provides a very efficient (FFT-based) implementation of this estimation procedure. It also includes a benchmark/demo for testing its optimality: a fBm signal is generated, corrupted by additive noise, and reconstructed from the noisy measurements with an oversampling factor of m.

Description

The available functions are:

• fsdemo.m: Demo/benchmark programme, described above.
• smoothspline.m: Function that returns samples of the smoothing spline for a given input sequence, regularization factor (lambda), spline order (gamma), and oversampling factor (m). See MATLAB help for details.
• fBmper.m: Generator of fractional (pseudo-)Brownian motion.
• fractsplineautocorr.m: Function for frequency domain computation of fractional spline autocorrelation.

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