A general model for deriving preference structures from data

In this paper we comment upon the integrated model for valued preferences introduced by Fodor, Ovchinnikov and Roubens. In particular, while on one hand we revise basic assumptions and point out their intuitive meaning, on the other hand we propose an alternative mathematical justification of such a model which allows not only a better understanding of the obtained results, but also a functional characterization of the whole family of solutions.     

A limit order book model for latency arbitrage

We consider a single security market based on a limit order book and two investors, with different speeds of trade execution. If the fast investor can preempt the slower investor, we show that this allows the fast trader to obtain risk free profits, but that these profits cannot be scaled. We derive the fast trader??s optimal behaviour when she has only distributional knowledge of the slow trader??s actions, with few restrictions on the possible prior distributions. We also consider the slower trader??s response to the presence of a fast trader in a market, and the effects of the introduction of a ??Tobin tax?? on financial transactions. We show that such a tax can lead to the elimination of profits from preemptive strategies. Consequently, a Tobin tax can both increase market efficiency and attract traders to a market.     

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.     

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.     

A hybrid observer for localization from noisy inertial data and sporadic position measurements

We propose an asymptotic position and speed observer for inertial navigation in the case where the position measurements are sporadic and affected by noise. We cast the problem in a hybrid dynamics framework where the continuous motion is affected by unknown continuous-time disturbances and the sporadic position measurements are affected by discrete-time noise. We show that the peculiar hybrid cascaded structure describing the estimation error dynamics is globally finite-gain exponentially ISS with gains depending intuitively on our tuning parameters. Experimental results, as well as the comparison with an Extended Kalman Filter (EKF), confirm the effectiveness of the proposed solution with an execution time two orders of magnitude faster and with a simplified observer tuning because our bounds are an explicit function of the observer tuning knobs.     

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.     

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.     

A note on transfer function model specification with noisy closed‐loop input data

We assume that the input in a transfer function noise model is determined by a proportional‐integral‐derivative controller that is subject to noise. We show how such a noisy feedback in the input series affects the standard Box–Jenkins method of specifying the transfer function model. Our findings have important implications for the analyst who uses the traditional Box–Jenkins approach. The analyst will be led to estimate an incorrect transfer function model if feedback (exact or noisy) is present. Copyright © 2009 John Wiley & Sons, Ltd.     

Competitive algorithms for the clustering of noisy data

In this paper, we consider the issue of clustering when outliers exist. The outlier set is defined as the complement of the data set. Following this concept, a specially designed fuzzy membership weighted objective function is proposed and the corresponding optimal membership is derived. Unlike the membership of fuzzy c-means, the derived fuzzy membership does not reduce with the increase of the cluster number. With the suitable redefinition of the distance metric, we demonstrate that the objective function could be used to extract c spherical shells. A hard clustering algorithm alleviating the prototype under-utilization problem is also derived. Artificially generated data are used for comparisons.     

A penalized nonparametric method for nonlinear constrained optimization based on noisy data

The objective of this study is to find a smooth function joining two points A and B with minimum length constrained to avoid fixed subsets. A penalized nonparametric method of finding the best path is proposed. The method is generalized to the situation where stochastic measurement errors are present. In this case, the proposed estimator is consistent, in the sense that as the number of observations increases the stochastic trajectory converges to the deterministic one. Two applications are immediate, searching the optimal path for an autonomous vehicle while avoiding all fixed obstacles between two points and flight planning to avoid threat or turbulence zones.     

A relaxation method for reconstructing objects from noisy X-rays

An algorithm is presented for estimating the density distribution in a cross section of an object from X-ray data, which in practice is unavoidably noisy. The data give rise to a large sparse system of inconsistent equations, not untypically 10⁵ equations with 10⁴ unknowns, with only about 1% of the coefficients non-zero. Using the physical interpretation of the equations, each equality can in principle be replaced by a pair of inequalities, giving us the limits within which we believe the sum must lie. An algorithm is proposed for solving this set of inequalities. The algorithm is basically a relaxation method. A finite convergence result is proved. In spite of the large size of the system, in the application area of interest practical solution on a computer is possible because of the simple geometry of the problem and the redundancy of equations obtained from nearby X-rays. The algorithm has been implemented, and is demonstrated by actual reconstructions.     

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.     

A square root interest rate model fitting discrete initial term structure data

This paper presents one-factor and multifactor versions of a term structure model in which the factor dynamics are given by Cox/Ingersoll/Ross (CIR) type ‘square root’ diffusions with piece wise constant parameters. The model is fitted to initial term structures given by a finite number of data points, interpolating endogenously. Closed form and near closed form solutions for a large class of fixed income derivatives are derived in terms of a compound noncentral chi-square distribution. An implementation of the model is discussed where the initial term structure of volatility is fitted via cap prices.     

Approximation by smoothing variational vector splines for noisy data

This paper addresses the problem of constructing some free-form curves and surfaces from given to different types of data: exact and noisy data. We extend the theory of D^m${\mathrm{D}}^m$-splines over a bounded domain for noisy data to the smoothing variational vector splines. Both results of convergence for respectively the exact and noisy data are established, as soon as some estimations of errors are given.     

Integral curves from noisy diffusion MRI data with closed-form uncertainty estimates

We present a method for estimating the trajectories of axon fibers through diffusion tensor MRI (DTI) data that provides theoretically rigorous estimates of trajectory uncertainty. We develop a three-step estimation procedure based on a kernel estimator for a tensor field based on the raw DTI measurements, followed by a plug-in estimator for the leading eigenvectors of the tensors, and a plug-in estimator for integral curves through the resulting vector field. The integral curve estimator is asymptotically normal; the covariance of the limiting Gaussian process allows us to construct confidence ellipsoids for fixed points along the curve. Complete trajectories of fibers are assembled by stopping integral curve tracing at locations with multiple viable leading eigenvector directions and tracing a new curve along each direction. Unlike probabilistic tractography approaches to this problem, we provide a rigorous, theoretically sound model of measurement uncertainty as it propagates from the raw MRI data, to the tensor field, to the vector field, to the integral curves. In addition, trajectory uncertainty is estimated in closed form while probabilistic tractography relies on sampling the space of tensors, vectors, or curves. We show that our estimator provides more realistic trajectory uncertainty estimates than a more simplified prior approach for closed-form trajectory uncertainty estimation due to Koltchinskii et al. (Ann Stat 35:1576–1607, 2007) and a popular probabilistic tractography method due to Behrens et al. (Magn Reson Med 50:1077–1088, 2003) using theory, simulation, and real DTI scans.     

Almost sure convergence of smoothingD m -splines for noisy data

Summary The purpose of this paper is to study the convergence of smoothingD ^m -splines relative to sets of data perturbed by a random noise. Conditions of almost sure convergence and error estimates are given.     

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.