Tomasz Galkowski

Nonparametric recovery of multivariate functions with applications to system identification

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

Kernel Estimation of Regression Functions in the Boundary Regions

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.

Nonparametric Function Fitting in the Presence of Nonstationary Noise

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 Extension of Regression Functions Outside Domain

The article refers to the problem of regression functions estimation in the points situated near the edges but outside of function 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(.) is a completely unknown function. In the literature the possible ways of finding unknown function are based on the algorithms derived from the Parzen kernel. These algorithms were also applied to estimation of the derivatives of unknown functions. The commonly known disadvantage of the kernel algorithms is that the error of estimation dramatically increases if the point of estimation x is approaching to the left or right bound of interval D. Algorithms on predicting values in the boundary region outside the function domain D are unknown for the author, so far.

Nonparametric Estimation of Edge Values of Regression Functions

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.

Iron Doped SBA-15 Mesoporous Silica Studied by Mössbauer Spectroscopy

Mesoporous silica SBA-15 containing propyl-iron-phosphonate groups were considered to confirm their molecular structure. To detect the iron-containing group configuration the Mossbauer spectroscopy was used. Both mesoporous silica SBA-15 containing propyl-iron-phosphonate groups and pure doping agent iron acetylacetate were investigated using Mossbauer spectroscopy. The parameters such as isomer shift, quadrupole splitting, and asymmetry in 57Fe Mossbauer spectra were analyzed. The differences in Mossbauer spectra were explained assuming different local surroundings of Fe nuclei. On this base we were able to conclude about activation of phosphonate units by iron ions and determinate the oxidation state of the metal ion. To examine bonding between iron atoms and phosphonic units the resonance Raman spectroscopy was applied. The density functional theory DFT approach was used to make adequate calculations. The distribution of active units inside silica matrix was estimated by comparison of calculated vibrational spectra with the experimental ones. Analysis of both Mossbauer and resonance Raman spectra seems to confirm the correctness of the synthesis procedure. Also EDX elemental analysis confirms our conclusions.

Orthogonal Series Estimation of Regression Functions in Nonstationary Conditions

The article concerns of the problem of regression functions estimation when the output is contaminated by additive nonstationary noise. We investigate the model \(y_i = R\left( {{\bf x _i}} \right) + Z _i ,\,i = 1,2, \ldots n\), where x i is assumed to be the set of deterministic inputs (d-dimensional vector), y i is the scalar, probabilistic outputs, and Z i is a measurement noise with zero mean and variance depending on n. \(R\left( . \right)\) is a completely unknown function. The problem of finding function \(R\left( . \right)\) may be solved by applying non-parametric methodology, for instance: algorithms based on the Parzen kernel or algorithms derived from orthogonal series. In this work we present the orthogonal series approach. The analysis has been made for some class of nonstationarity. We present the conditions of convergence of the estimation algorithm for the variance of noise growing up when number of observations is tending to infinity. The results of numerical simulations are presented.

Improvement of the Multiple-View Learning Based on the Self-Organizing Maps

Big data sets and variety of data types lead to new types of problems in modern intelligent data analysis. This requires the development of new techniques and models. One of the important subjects is to reveal and indicate heterogeneous of non-trivial features of a large database. Original techniques of modelling, data mining, pattern recognition, machine learning in such fields like commercial behaviour of Internet users, social networks analysis, management and investigation of various databases in static or dynamic states have been recently investigated. Many techniques discovering hidden structures in the data set like clustering and projection of data from high-dimensional spaces have been developed. In this paper we have proposed a model for multiple view unsupervised clustering based on Kohonen self-organizing-map method.

The Novel Method of the Estimation of the Fourier Transform Based on Noisy Measurements

This article refers to the problem of the analysis of spectrum of signals observed in the presence of noise. We propose a new concept of estimation of the frequency content in the signal. The method is derived from the nonparametric methodology of function estimation. We refer to the model of the system \(y_i = R\left( {x_i } \right) + \epsilon _i ,\,i = 1,2, \ldots n\), 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. In this paper we are interested in a question about frequency spectrum of unknown function. Finding of unknown function in the model could be realized using algorithms based on the Parzen kernel. The alternative approach is based on the orthogonal series expansions. Nonparametric methodology could also be used in the task of implicit estimation of its spectrum. The main aim of this paper is to propose an original integral version of nonparametric estimation of spectrum based on trigonometric series - referring to the classic Fourier transform. The results of numerical experiments are presented.

The Concept of Molecular Neurons

The paper concerns the main element of the molecular neural network - the Molecular Neuron (MN). Molecular Neural Network idea has been introduced in our previous articles. Here we present the structure of the Molecular Neuron element in micro and nanoscale. We have obtained MN in hexagonal layout in the form of the thin film. In this paper we have described self-assembly mechanism leading to the NMs layout. Also physical properties of the MNs layer have been shown.