Fixed design regression for time series: asymptotic normality

Consistent nonparametric multiple regression: the fixed design case

A nonparametric algorithm, based on the multiple Fejer's sum, for fitting of multivariate functions is proposed. The mean square error convergence and the speed of convergence are investigated.

Nonparametric fitting of multivariate functions

A nonparametric algorithm, based on multiple Fourier series, for identification of MISO (multidimensional-input-single-output) systems is investigated. It is shown that the algorithm converges for a wide class of noise. >

Identification of MISO nonlinear regressions in the presence of a wide class of disturbances

Three nonparametric procedures for the extraction of nonlinear regressions from noisy data are proposed. The procedures are based on the Dirichlet, Fejer, and de la Vallee Poussin multiple kernels. Convergence properties are investigated. In particular, it is shown that the algorithms are convergent in the mean-integrated-square-error sense. The appropriate theorem establishes a relation between the order of kernels and the number of observations. Special attention is focused on the two-dimensional case. It is proved that the procedures attain the optimal rate of convergence, which cannot be exceeded by any other nonparametric algorithm. >

Multiple Fourier series procedures for extraction of nonlinear regressions from noisy data

The methodology for nonparametric fitting of multivariate functions is proposed. The mean-square error convergence and the strong convergence are proved.

Nonparametric recovery of multivariate functions with applications to system identification

The article refers to the problem of regression functions estimation in the points near the edges of their domain. We investigate the model \(y_i = R\left( {x_i } \right) + \epsilon _i ,\,i = 1,2, \ldots n\), where x i is assumed to be the set of deterministic inputs, x i ∈ D, y i is the set of probabilistic outputs, and e i is a measurement noise with zero mean and bounded variance. \(R\left( . \right)\) is a completely unknown function. The possible clue of finding unknown function is to apply the algorithms based on Parzen kernel [5], [12]. The commonly known inconvenience of these algorithms is that the error of estimation dramatically increases if the point of estimation x is coming up to the left or right bound of interval D.

Kernel Estimation of Regression Functions in the Boundary Regions

Abstract Nowadays, unprecedented amounts of heterogeneous data collections are stored, processed and transmitted via the Internet. In data analysis one of the most important problems is to verify whether data observed or/and collected in time are genuine and stationary, i.e. the information sources did not change their characteristics. There is a variety of data types: texts, images, audio or video files or streams, metadata descriptions, thereby ordinary numbers. All of them changes in many ways. If the change happens the next question is what is the essence of this change and when and where the change has occurred. The main focus of this paper is detection of change and classification of its type. Many algorithms have been proposed to detect abnormalities and deviations in the data. In this paper we propose a new approach for abrupt changes detection based on the Parzen kernel estimation of the partial derivatives of the multivariate regression functions in presence of probabilistic noise. The proposed change detection algorithm is applied to oneand two-dimensional patterns to detect the abrupt changes.

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A New Approach to Detection of Changes in Multidimensional Patterns

The article refers to the problem of regression functions estimation in the presence of nonstationary noise. We investigate the model \(y_i = R\left( {{\bf x _i}} \right) + \epsilon _i ,\,i = 1,2, \ldots n\), where x i is assumed to be the d-dimensional vector, set of deterministic inputs, x i ∈ S d, y i is the scalar, set of probabilistic outputs, and e i is a measurement noise with zero mean and variance depending on n. \(R\left( . \right)\) is a completely unknown function. One of the possible solutions of finding function \(R\left( . \right)\) is to apply non-parametric methodology - algorithms based on the Parzen kernel or algorithms derived from orthogonal series. The novel result of this article is the analysis of convergence for some class of nonstationarity. We present the conditions when the algorithm of estimation is convergent even when the variance of noise is divergent with number of observations tending to infinity. The results of numerical experiments are presented.

Nonparametric Function Fitting in the Presence of Nonstationary Noise

In this article we investigate the problem of regression functions estimation in the edges points of their domain. We refer to the model \(y_i = R\left( {x_i } \right) + \epsilon _i ,\,i = 1,2, \ldots n\), where \(x_i\) is assumed to be the set of deterministic inputs, \(x_i \in D\), \(y_i\) is the set of probabilistic outputs, and \(\epsilon _i\) is a measurement noise with zero mean and bounded variance. R(.) is a completely unknown function. The possible solution of finding unknown function is to apply the algorithms based on the Parzen kernel [13, 31]. The commonly known drawback of these algorithms is that the error of estimation dramatically increases if the point of estimation x is drifting to the left or right bound of interval D. This fact makes it impossible to estimate functions exactly in edge values of domain.

Tomasz Galkowski

Papers

Nonparametric recovery of multivariate functions with applications to system identification

Kernel Estimation of Regression Functions in the Boundary Regions

A New Approach to Detection of Changes in Multidimensional Patterns

Nonparametric Function Fitting in the Presence of Nonstationary Noise

Nonparametric Estimation of Edge Values of Regression Functions