Spline approximation of a random process with singularity

Let a continuous random process $X$ defined on $[0,1]$ be $(m+\beta)$-smooth, $0\le m, 00$ and have an isolated singularity point at $t=0$. In addition, let $X$ be locally like a $m$-fold integrated $\beta$-fractional Brownian motion for all non-singular points. We consider approximation of $X$ by piecewise Hermite interpolation splines with $n$ free knots (i.e., a sampling design, a mesh). The approximation performance is measured by mean errors (e.g., integrated or maximal quadratic mean errors). We construct a sequence of sampling designs with asymptotic approximation rate $n^{-(m+\beta)}$ for the whole interval.Comment: 16 pages, 2 figure typos and references corrected, revised classes definition, results unchange

Free-Knot Spline Approximation of Stochastic Processes

We study optimal approximation of stochastic processes by polynomial splines with free knots. The number of free knots is either a priori fixed or may depend on the particular trajectory. For the $s$-fold integrated Wiener process as well as for scalar diffusion processes we determine the asymptotic behavior of the average $L_p$-distance to the splines spaces, as the (expected) number $k$ of free knots tends to infinity.Comment: 23 page

A Functional Wavelet-Kernel Approach for Continuous-time Prediction

We consider the prediction problem of a continuous-time stochastic process on an entire time-interval in terms of its recent past. The approach we adopt is based on functional kernel nonparametric regression estimation techniques where observations are segments of the observed process considered as curves. These curves are assumed to lie within a space of possibly inhomogeneous functions, and the discretized times series dataset consists of a relatively small, compared to the number of segments, number of measurements made at regular times. We thus consider only the case where an asymptotically non-increasing number of measurements is available for each portion of the times series. We estimate conditional expectations using appropriate wavelet decompositions of the segmented sample paths. A notion of similarity, based on wavelet decompositions, is used in order to calibrate the prediction. Asymptotic properties when the number of segments grows to infinity are investigated under mild conditions, and a nonparametric resampling procedure is used to generate, in a flexible way, valid asymptotic pointwise confidence intervals for the predicted trajectories. We illustrate the usefulness of the proposed functional wavelet-kernel methodology in finite sample situations by means of three real-life datasets that were collected from different arenas

Convergence of Fourier-Wavelet models for Gaussian random processes

Mean square convergence and convergence in probability of Fourier-Wavelet Models (FWM) of stationary Gaussian Random processes in the metric of Banach space of continuously differentiable functions and in Sobolev space are studied. Sufficient conditions for the convergence formulated in the frame of spectral functions are given. It is shown that the given rates of convergence of FWM in the mean square obtained in the Nikolskiui-Besov classes cannot be improved

Convergence of Fourier-wavelet models for Gaussian random processes

Mean square convergence and convergence in probability of Fourier-Wavelet Models (FWM) of stationary Gaussian Random processes in the metric of Banach space of continuously differentiable functions and in Sobolev space are studied. Sufficient conditions for the convergence formulated in the frame of spectral functions are given. It is shown that the given rates of convergence of FWM in the mean square obtained in the Nikolski\u{i}-Besov classes cannot be improved

Local Mixture Model in Hilbert Space

In this thesis, we study local mixture models with a Hilbert space structure. First, we consider the fibre bundle structure of local mixture models in a Hilbert space. Next, the spectral decomposition is introduced in order to construct local mixture models. We analyze the approximation error asymptotically in the Hilbert space. After that, we will discuss the convexity structure of local mixture models. There are two forms of convexity conditions to consider, first due to positivity in the $-1$-affine structure and the second by points having to lie inside the convex hull of a parametric family. It is shown that the set of mixture densities is located inside the intersection of the sets defined by these two convexities. Finally, we discuss the impact of the approximation error in the Hilbert space when the domain of mixing variable changes