Non-parametric estimation of the diffusion coefficient from noisy data

We consider a diffusion process (X _t ) _t????0, with drift b(x) and diffusion coefficient ??(x). At discrete times t _k ?=?k ?? for k from 1 to M, we observe noisy data of the sample path, ${Y_{k\delta}=X_{k\delta}+\varepsilon_{k}}$ . The random variables ${\left(\varepsilon_{k}\right)}$ are i.i.d, centred and independent of (X _t ). The process (X _t ) _t????0 is assumed to be strictly stationary, ??-mixing and ergodic. In order to reduce the noise effect, we split data into groups of equal size p and build empirical means. The group size p is chosen such that ???=?p ?? is small whereas M ?? is large. Then, the diffusion coefficient ?? ² is estimated in a compact set A in a non-parametric way by a penalized least squares approach and the risk of the resulting adaptive estimator is bounded. We provide several examples of diffusions satisfying our assumptions and we carry out various simulations. Our simulation results illustrate the theoretical properties of our estimators.     

Jump-preserving surface reconstruction from noisy data

A new local smoothing procedure is suggested for jump-preserving surface reconstruction from noisy data. In a neighborhood of a given point in the design space, a plane is fitted by local linear kernel smoothing, giving the conventional local linear kernel estimator of the surface at the point. The neighborhood is then divided into two parts by a line passing through the given point and perpendicular to the gradient direction of the fitted plane. In the two parts, two half planes are fitted, respectively, by local linear kernel smoothing, providing two one-sided estimators of the surface at the given point. Our surface reconstruction procedure then proceeds in the following two steps. First, the fitted surface is defined by one of the three estimators, i.e., the conventional estimator and the two one-sided estimators, depending on the weighted residual means of squares of the fitted planes. The fitted surface of this step preserves the jumps well, but it is a bit noisy, compared to the conventional local linear kernel estimator. Second, the estimated surface values at the original design points obtained in the first step are used as new data, and the above procedure is applied to this data in the same way except that one of the three estimators is selected based on their estimated variances. Theoretical justification and numerical examples show that the fitted surface of the second step preserves jumps well and also removes noise efficiently. Besides two window widths, this procedure does not introduce other parameters. Its surface estimator has an explicit formula. All these features make it convenient to use and simple to compute.     

Smoothing noisy data with spline functions

It is shown how to choose the smoothing parameter when a smoothing periodic spline of degree 2m?1 is used to reconstruct a smooth periodic curve from noisy ordinate data. The noise is assumed “white”, and the true curve is assumed to be in the Sobolev spaceW ₂ ^(2m) of periodic functions with absolutely continuousv-th derivative,v=0, 1, ..., 2m?1 and square integrable 2m-th derivative. The criteria is minimum expected square error, averaged over the data points. The dependency of the optimum smoothing parameter on the sample size, the noise variance, and the smoothness of the true curve is found explicitly.     

Smoothing noisy data with spline functions

Summary A procedure for calculating the trace of the influence matrix associated with a polynomial smoothing spline of degree2m–1 fitted ton distinct, not necessarily equally spaced or uniformly weighted, data points is presented. The procedure requires orderm ² n operations and therefore permits efficient orderm ² n calculation of statistics associated with a polynomial smoothing spline, including the generalized cross validation. The method is a significant improvement over an existing method which requires ordern ³ operations.     

Error bounds for derivative estimates based on spline smoothing of exact or noisy data

Estimates are found for the L₂ error in approximating the jth derivative of a given smooth function f by the corresponding derivative of the 2mth order smoothing spline based on an n-point sample from the function. The results cover both the case of an exact sample from f and the case when the sample is subject to some random noise. In the noisy case, the estimates are for the expected value of the approximation error. These bounds show that, even in the presence of noise, the derivatives of the smoothing splines of order less than m can be expected to converge to those of f as the number of (uniform) sample points increases, and the smoothing parameter approaches zero at a rate appropriately related to m, n, and the order of differentiability of f.     

Generalization bounds for function approximation from scattered noisy data

We consider the problem of approximating functions from scattered data using linear superpositions of non-linearly parameterized functions. We show how the total error (generalization error) can be decomposed into two parts: an approximation part that is due to the finite number of parameters of the approximation scheme used; and an estimation part that is due to the finite number of data available. We bound each of these two parts under certain assumptions and prove a general bound for a class of approximation schemes that include radial basis functions and multilayer perceptrons. This revised version was published online in June 2006 with corrections to the Cover Date.     

Nonparametric reconstruction of a multifractal function from noisy data

We estimate a real-valued function f of d variables, subject to additive Gaussian perturbation at noise level ${\varepsilon > 0}$ , under L ^π -loss, for π ≥ 1. The main novelty is that f can have an extremely varying local smoothness, exhibiting a so-called multifractal behaviour. The results of Jaffard on the Frisch–Parisi conjecture suggest to link the singularity spectrum of f to Besov properties of the signal that can be handled by wavelet thresholding for denoising purposes. We prove that the optimal (minimax) rate of estimation of multifractal functions with singularity spectrum d(H) has explicit representation ${\varepsilon^{2v(d({\bullet}),\pi)}}$ , with $$ v(d({\bullet}),\pi)=\min_{H}\frac{H+\left(d-d(H)\right)/\pi}{2H+d}.$$ The minimum is taken over a specific domain and the rate is corrected by logarithmic factors in some cases. In particular, the usual rate ${\varepsilon^{2s/(2s+d)}}$ is retrieved for monofractal functions (with spectrum reduced to a single value s) irrespectively of π. More interestingly, the sparse case of estimation over single Besov balls has a new interpretation in terms of multifractal analysis.     

Gradient estimates for diffusion semigroups with singular coefficients

Uniform gradient estimates are derived for diffusion semigroups, possibly with potential, generated by second order elliptic operators having irregular and unbounded coefficients. We first consider the Rd-case, by using the coupling method. Due to the singularity of the coefficients, the coupling process we construct is not strongly Markovian, so that additional difficulties arise in the study. Then, more generally, we treat the case of a possibly unbounded smooth domain of Rd with Dirichlet boundary conditions. We stress that the resulting estimates are new even in the Rd-case and that the coefficients can be Hölder continuous. Our results also imply a new Liouville theorem for space-time bounded harmonic functions with respect to the underlying diffusion semigroup.     

Irregularity detection from noisy data in one and two dimensions

In this work we consider the problem of detecting the irregularities of univariate functions from noisy data and its extension to bivariate functions which present lines of points of irregularity. This revised version was published online in August 2006 with corrections to the Cover Date.     

Restriction estimates for space curves with respect to general measures

In this paper we consider adjoint restriction estimates for space curves with respect to general measures and obtain optimal estimates when the curves satisfy a finite type condition. The argument here is new in that it doesn?t rely on the offspring curve method, which has been extensively used in the previous works. Our work was inspired by the recent argument due to Bourgain and Guth which was used to deduce linear restriction estimates from multilinear estimates for hypersurfaces.     

A filtered no arbitrage model for term structures from noisy data

We consider an affine term structure model of interest rates, where the factors satisfy a linear diffusion equation. We assume that the information available to an agent comes from observing the yields of a finite number of traded bonds and that this information is not sufficient to reconstruct exactly the factors. We derive a method to obtain arbitrage-free prices of illiquid or non traded bonds that are compatible with the available incomplete information. The method is based on an application of the Kalman filter for linear Gaussian systems.     

Training neural networks with noisy data as an ill-posed problem

This paper is devoted to the analysis of network approximation in the framework of approximation and regularization theory. It is shown that training neural networks and similar network approximation techniques are equivalent to least-squares collocation for a corresponding integral equation with mollified data.Results about convergence and convergence rates for exact data are derived based upon well-known convergence results about least-squares collocation. Finally, the stability properties with respect to errors in the data are examined and stability bounds are obtained, which yield rules for the choice of the number of network elements.     

Cost efficiency measures in data envelopment analysis with data uncertainty

This paper extends the classical cost efficiency (CE) models to include data uncertainty. We believe that many research situations are best described by the intermediate case, where some uncertain input and output data are available. In such cases, the classical cost efficiency models cannot be used, because input and output data appear in the form of ranges. When the data are imprecise in the form of ranges, the cost efficiency measure calculated from the data should be uncertain as well. So, in the current paper, we develop a method for the estimation of upper and lower bounds for the cost efficiency measure in situations of uncertain input and output data. Also, we develop the theory of efficiency measurement so as to accommodate incomplete price information by deriving upper and lower bounds for the cost efficiency measure. The practical application of these bounds is illustrated by a numerical example.     

Rapid parameter identification of linear time-delay system from noisy frequency domain data

The technique to identify the system parameters thereof has attracted extensive research interest, since knowing the parameters would enable effective system control strategy and accurate response prediction. In this paper, a novel approach is developed to identify the parameters of the linear time-delay differential system by analyzing the complex system response in the frequency domain. Firstly, the complex frequency response of the time-delay system is expressed as a function of physical parameters and time-delay parameters, forming a typical optimization problem. Subsequently, the sensitivities with respect to the unknown parameters are derived. A novel sensitivity-based algorithm is adopted in the identification procedure. Trust-region constraint is implemented and hence tackled by Tikhonov regularization, which effectively enhances the efficiency of the algorithm. The feasibility and robustness of the identification procedure are evaluated by identifying the parameters of two numerical time-delay systems and an experimental case.     

Regularization of some linear ill-posed problems with discretized random noisy data

For linear statistical ill-posed problems in Hilbert spaces we introduce an adaptive procedure to recover the unknown solution from indirect discrete and noisy data. This procedure is shown to be order optimal for a large class of problems. Smoothness of the solution is measured in terms of general source conditions. The concept of operator monotone functions turns out to be an important tool for the analysis.

     

Difference approximations to interface curves for nonlinear diffusion equations with absorption

We consider some nonlinear reaction-diffusion equations with extinction phenomena in finite time. For the solution v, there appear interface curves between v > 0 and v = 0. We propose difference approximations to interface curves, and prove the convergence to exact interface curves.