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1
Content available remote

On least squares estimation of Fourier coefficients and of the regression function

100%
Popiński W.
Applicationes Mathematicae
|
1993-1995
|
tom 22
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nr 1
91-102
EN
The problem of nonparametric function fitting with the observation model $y_i = f(x_i) + η_i$, i=1,...,n, is considered, where $η_i$ are independent random variables with zero mean value and finite variance, and $x_i \in [a,b] \subset \R^1$, i=1,...,n, form a random sample from a distribution with density $ϱ \in L^1[a,b]$ and are independent of the errors $η_i$, i=1,...,n. The asymptotic properties of the estimator $\widehat{f}_{N(n)}(x) = \sum_{k=1}^{N(n)} \widehat{c}_ke_k(x)$ for $f \in L^2[a,b]$ and $\widehat{c}^{N(n)}=( \widehat{c}_1,..., \widehat{c}_{N(n)})^T$ obtained by the least squares method as well as the limits in probability of the estimators $\widehat{c}_k$, k=1,...,N, for fixed N, are studied in the case when the functions $e_k$, k=1,2,..., forming a complete orthonormal system in $L^2\[a,b\]$ are analytic.
2
Content available remote

A note on orthogonal series regression function estimators

100%
Popiński W.
Applicationes Mathematicae
|
1999
|
tom 26
|
nr 3
281-291
EN
The problem of nonparametric estimation of the regression function f(x) = E(Y | X=x) using the orthonormal system of trigonometric functions or Legendre polynomials $e_k$, k=0,1,2,..., is considered in the case where a sample of i.i.d. copies $(X_i,Y_i)$, i=1,...,n, of the random variable (X,Y) is available and the marginal distribution of X has density ϱ ∈ $L^1$[a,b]. The constructed estimators are of the form $\widehat f_n(x) = \sum_{k=0}^{N(n)}\widehat c_ke_k(x)$, where the coefficients $\widehat c_0,\widehat c_1,...,\widehat c_N$ are determined by minimizing the empirical risk $n^{-1}\sum_{i=1}^n(Y_i - \sum_{k=0}^Nc_ke_k(X_i))^2$. Sufficient conditions for consistency of the estimators in the sense of the errors $E_X\vert f(X)-\widehat f_n(X)\vert^2$ and $n^{-1}\sum_{i=1}^nE(f(X_i)-\widehat f_n(X_i))^2$ are obtained.
3
Content available remote

Orthogonal series estimation of band-limited regression functions

100%
Popiński W.
Applicationes Mathematicae
|
2014
|
tom 41
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nr 1
51-65
EN
The problem of nonparametric function fitting using the complete orthogonal system of Whittaker cardinal functions $s_k$, k = 0,±1,..., for the observation model $y_j = f(u_j) + η_j$, j = 1,...,n, is considered, where f ∈ L²(ℝ) ∩ BL(Ω) for Ω > 0 is a band-limited function, $u_j$ are independent random variables uniformly distributed in the observation interval [-T,T], $η_j$ are uncorrelated or correlated random variables with zero mean value and finite variance, independent of the observation points. Conditions for convergence and convergence rates of the integrated mean-square error E||f-f̂ₙ||² and the pointwise mean-square error E(f(x)-f̂ₙ(x))² of the estimator $f̂ₙ(x) = ∑_{k=-N(n)}^{N(n)} ĉ_k s_k(x)$ with coefficients $ĉ_k$, k = -N(n),...,N(n), obtained by the Monte Carlo method are studied.
4
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Convergence rates of orthogonal series regression estimators

100%
Popiński W.
Applicationes Mathematicae
|
2000
|
tom 27
|
nr 4
445-454
EN
General conditions for convergence rates of nonparametric orthogonal series estimators of the regression function f(x)=E(Y | X = x) are considered. The estimators are obtained by the least squares method on the basis of a random observation sample (Y_i,X_i), i=1,...,n, where $X_i ∈ A ⊂ ℝ^d$ have marginal distribution with density $ϱ ∈ L^1(A)$ and Var( Y | X = x) is bounded on A. Convergence rates of the errors $E_X(f(X)-\widehat f_N(X))^2$ and $\Vert f-\widehat f_N\Vert_∞$ for the estimator $\widehat f_N(x) = \sum_{k=1}^N\widehat c_ke_k(x)$, constructed using an orthonormal system $e_k$, k=1,2,..., in $L^2(A)$ are obtained.
5
Content available remote

Consistency of trigonometric and polynomial regression estimators

100%
Popiński W.
Applicationes Mathematicae
|
1998-1999
|
tom 25
|
nr 1
73-83
EN
The problem of nonparametric regression function estimation is considered using the complete orthonormal system of trigonometric functions or Legendre polynomials $e_k$, k=0,1,..., for the observation model $y_i = f(x_i) + η_i $, i=1,...,n, where the $η_i$ are independent random variables with zero mean value and finite variance, and the observation points $x_i\in[a,b]$, i=1,...,n, form a random sample from a distribution with density $ϱ\in L^1[a,b]$. Sufficient and necessary conditions are obtained for consistency in the sense of the errors $\Vert f-\widehat f_N\Vert, \vert f(x)-\widehatf_N(x)\vert$, $x\in[a,b]$, and $E\Vert f-\widehatf_N\Vert^2$ of the projection estimator $\widehat f_N(x) = \sum_{k=0}^N\widehat{c}_ke_k(x)$ for $\widehat{c}_0,\widehat{c}_1,\ldots,\widehat{c}_N$ determined by the least squares method and $f\in L^2[a,b]$.
6
Content available remote

Least-squares trigonometric regression estimation

100%
Popiński W.
Applicationes Mathematicae
|
1999
|
tom 26
|
nr 2
121-131
EN
The problem of nonparametric function fitting using the complete orthogonal system of trigonometric functions $e_k$, k=0,1,2,..., for the observation model $y_i = f(x_{in}) + η_i$, i=1,...,n, is considered, where $η_i$ are uncorrelated random variables with zero mean value and finite variance, and the observation points $x_{in} ∈ [0,2π]$, i=1,...,n, are equidistant. Conditions for convergence of the mean-square prediction error $(1/n)\sum_{i=1}^n E(f(x_{in})-\widehat f_{N(n)}(x_{in}))^2$, the integrated mean-square error $E ‖f-\widehat f_{N(n)}‖^2$ and the pointwise mean-square error $E(f(x)-\widehatf_{N(n)}(x))^2$ of the estimator $\widehat f_{N(n)}(x) = \sum_{k=0}^{N(n)} \widehat c_k e_k(x)$ for f ∈ C[0,2π] and $\widehat c_0,\widehat c_1,...,\widehat c_{N(n)}$ obtained by the least squares method are studied.
7
Content available remote

On discrete Fourier spectrum of a harmonic with random frequency modulation

100%
Popiński W.
Applicationes Mathematicae
|
2013
|
tom 40
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nr 1
99-108
EN
Asymptotic properties of the Discrete Fourier Transform spectrum of a complex monochromatic oscillation with frequency randomly distorted at the observation times t=0,1,..., n-1 by a series of independent and identically distributed fluctuations is investigated. It is proved that the second moments of the spectrum at the discrete Fourier frequencies converge uniformly to zero as n → ∞ for certain frequency fluctuation distributions. The observed effect occurs even for frequency fluctuations with magnitude arbitrarily small in comparison to the original oscillation frequency.
8
Content available remote

On discrete Fourier analysis of amplitude and phase modulated signals

100%
Popiński W.
Applicationes Mathematicae
|
2012
|
tom 39
|
nr 1
57-69
EN
In this work the problem of characterization of the Discrete Fourier Transform (DFT) spectrum of an original complex-valued signal $o_t$, t=0,1,...,n-1, modulated by random fluctuations of its amplitude and/or phase is investigated. It is assumed that the amplitude and/or phase of the signal at discrete times of observation are distorted by realizations of uncorrelated random variables or randomly permuted sequences of complex numbers. We derive the expected values and bounds on the variances of such distorted signal DFT spectra. It is shown that the modulation considered in general entails changes in the amplitude and/or phase of the DFT spectra expected values, which together with imposed random deviations with finite variances can vary the amplitudes of peaks existing in the original signal spectrum, and consequently similarity to the original signal spectrum can be significantly blurred.
9
Content available remote

On Fourier coefficient estimators consistent in the mean-square sense

100%
Popiński W.
Applicationes Mathematicae
|
1993-1995
|
tom 22
|
nr 2
275-284
EN
The properties of two recursive estimators of the Fourier coefficients of a regression function $f \in L^2[a,b]$ with respect to a complete orthonormal system of bounded functions (e_k) , k=1,2,..., are considered in the case of the observation model $y_i = f(x_i) + η_i$, i=1,...,n , where $η_i$ are independent random variables with zero mean and finite variance, $x_i \in [a,b] \subset {\sym R}^1$, i=1,...,n, form a random sample from a distribution with density ϱ =1/(b-a) (uniform distribution) and are independent of the errors $η_i$, i=1,...,n . Unbiasedness and mean-square consistency of the examined estimators are proved and their mean-square errors are compared.
10
Content available remote

Orthogonal series regression estimators for an irregularly spaced design

100%
Popiński W.
Applicationes Mathematicae
|
2000
|
tom 27
|
nr 3
309-318
EN
Nonparametric orthogonal series regression function estimation is investigated in the case of a fixed point design where the observation points are irregularly spaced in a finite interval [a,b]i ⊂ ℝ. Convergence rates for the integrated mean-square error and pointwise mean-square error are obtained in the case of estimators constructed using the Legendre polynomials and Haar functions for regression functions satisfying the Lipschitz condition.
11
Content available remote

Orthogonal series regression estimation under long-range dependent errors

100%
Popiński W.
Applicationes Mathematicae
|
2001
|
tom 28
|
nr 4
457-466
EN
This paper is concerned with general conditions for convergence rates of nonparametric orthogonal series estimators of the regression function. The estimators are obtained by the least squares method on the basis of an observation sample $Y_i = f(X_i) + η_i$, i=1,...,n, where $X_i ∈ A ⊂ ℝ^d$ are independently chosen from a distribution with density ϱ ∈ L¹(A) and $η_i$ are zero mean stationary errors with long-range dependence. Convergence rates of the error $n^{-1} ∑_{i=1}^n (f(X_i)-f̂_N(X_i))²$ for the estimator $f̂_N(x) = ∑_{k=1}^N ĉ_k e_k(x)$, constructed using an orthonormal system $e_k$, k=1,2,..., in L²(A), are obtained.
12
Content available remote

On orthogonal series estimation of bounded regression functions

100%
Popiński W.
Applicationes Mathematicae
|
2001
|
tom 28
|
nr 3
257-270
EN
The problem of nonparametric estimation of a bounded regression function $f ∈ L²([a,b]^d)$, [a,b] ⊂ ℝ, d ≥ 1, using an orthonormal system of functions $e_k$, k=1,2,..., is considered in the case when the observations follow the model $Y_i = f(X_i) + η_i$, i=1,...,n, where $X_i$ and $η_i$ are i.i.d. copies of independent random variables X and η, respectively, the distribution of X has density ϱ, and η has mean zero and finite variance. The estimators are constructed by proper truncation of the function $f̂ₙ(x) = ∑_{k=1}^{N(n)} ĉ_k e_k(x)$, where the coefficients $ĉ₁,...,ĉ_{N(n)}$ are determined by minimizing the empirical risk $n^{-1} ∑_{i=1}^n (Y_i - ∑_{k=1}^{N(n)} c_k e_k(X_i))²$. Sufficient conditions for convergence rates of the generalization error $E_X | f(X)-f̂ₙ(X)|²$ are obtained.
13
Content available remote

Wavelet transform for time-frequency representation and filtration of discrete signals

100%
Popiński W.
Applicationes Mathematicae
|
1995-1996
|
tom 23
|
nr 4
433-448
EN
A method to analyse and filter real-valued discrete signals of finite duration s(n), n=0,1,...,N-1, where $N=2^p$, p>0, by means of time-frequency representation is presented. This is achieved by defining an invertible discrete transform representing a signal either in the time or in the time-frequency domain, which is based on decomposition of a signal with respect to a system of basic orthonormal discrete wavelet functions. Such discrete wavelet functions are defined using the Meyer generating wavelet spectrum and the classical discrete Fourier transform between the time and the frequency domains.
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