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11
Wiener–Kolmogorov Filtering, FrequencySelective Filtering and Polynomial Regression, Econometric Theory
, 2006
"... Adaptations of the classical Wiener–Kolmogorov filters are described that enable them to be applied to short nonstationary sequences. Alternative filtering methods that operate in the time domain and the frequency domain are described. The frequencydomain methods have the advantage of allowing comp ..."
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Cited by 7 (4 self)
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Adaptations of the classical Wiener–Kolmogorov filters are described that enable them to be applied to short nonstationary sequences. Alternative filtering methods that operate in the time domain and the frequency domain are described. The frequencydomain methods have the advantage of allowing components of the data to be separated along sharp dividing lines in the frequency domain, without incurring any leakage. The paper contains a novel treatment of the startup problem that affects the filtering of trended data sequences.
Improved FrequencySelective Filters
 Computational Statistics and Data Analysis
, 2003
"... A filtered data sequence can be obtained by multiplying the Fourier ordinates of the data by the ordinates of the frequency response of the filter and by applying the inverse Fourier transform to carry the product back to the time domain. Using this technique, it is possible, within the constraints ..."
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Cited by 2 (1 self)
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A filtered data sequence can be obtained by multiplying the Fourier ordinates of the data by the ordinates of the frequency response of the filter and by applying the inverse Fourier transform to carry the product back to the time domain. Using this technique, it is possible, within the constraints of a finite sample, to design an ideal frequencyselective filter that will preserve all elements within a specified range of frequencies and that will remove all elements outside it. Approximations to ideal filters that are implemented in the time domain are commonly based on truncated versions of the infinite sequences of coefficients derived from the Fourier transforms of rectangular frequency response functions. An alternative to truncating an infinite sequence of coefficients is to wrap it around a circle of a circumference equal in length to the data sequence and to add the overlying coefficients. The coefficients of the wrapped filter can also be obtained by applying a discrete Fourier transform to a set of ordinates sampled from the frequency response function. Applying the coefficients to the data via circular convolution produces results that are identical to those obtained by a multiplication in the frequency domain, which constitutes a more efficient approach.
Realisations of FiniteSample FrequencySelective Filters
"... A filtered data sequence can be obtained by multiplying the Fourier ordinates of the data by the ordinates of the frequency response of the filter and by applying the inverse Fourier transform to carry the product back to the time domain. Using this technique, it is possible, within the constraints ..."
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Cited by 2 (1 self)
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A filtered data sequence can be obtained by multiplying the Fourier ordinates of the data by the ordinates of the frequency response of the filter and by applying the inverse Fourier transform to carry the product back to the time domain. Using this technique, it is possible, within the constraints of a finite sample, to design an ideal frequencyselective filter that will preserve all elements within a specified range of frequencies and that will remove all elements outside it. Approximations to ideal filters that are implemented in the time domain are commonly based on truncated versions of the infinite sequences of coefficients derived from the Fourier transforms of rectangular frequency response functions. An alternative to truncating an infinite sequence of coefficients is to wrap it around a circle of a circumference equal in length to the data sequence and to add the overlying coefficients. The coefficients of the wrapped filter can also be obtained by applying a discrete Fourier transform to a set of ordinates sampled from the frequency response function. Applying the coefficients to the data via circular convolution produces results that are identical to those obtained by a multiplication in the frequency domain, which constitutes a more efficient approach. Key words: Signal extraction, Linear filtering, Frequencydomain analysis 1 Introduction: The Problem
The Identical Estimates of Spectral Norms for Circulant Matrices with Binomial Coefficients Combined with Fibonacci Numbers and Lucas Numbers Entries
"... Improved estimates for spectral norms of circulant matrices are investigated, and the entries are binomial coefficients combined with either Fibonacci numbers or Lucas numbers. Employing the properties of given circulant matrices, this paper improves the inequalities for their spectral norms, and g ..."
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Cited by 1 (0 self)
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Improved estimates for spectral norms of circulant matrices are investigated, and the entries are binomial coefficients combined with either Fibonacci numbers or Lucas numbers. Employing the properties of given circulant matrices, this paper improves the inequalities for their spectral norms, and gets corresponding identities of spectral norms. Moreover, by some wellknown identities, the explicit identities for spectral norms are obtained. Some numerical tests are listed to verify the results.
Bandlimited stochastic Processes in discrete and continuous time
 Studies in Nonlinear Dynamics & Econometrics 16, Iss. 1, Paper
, 2012
"... In the theory of stochastic differential equations, it is commonly assumed that the forcing function is a Wiener process. Such a process has an infinite bandwidth in the frequency domain. In practice, however, all stochastic processes have a limited bandwidth. A theory of bandlimited linear stocha ..."
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In the theory of stochastic differential equations, it is commonly assumed that the forcing function is a Wiener process. Such a process has an infinite bandwidth in the frequency domain. In practice, however, all stochastic processes have a limited bandwidth. A theory of bandlimited linear stochastic processes is described that reflects this reality, and it is shown how the corresponding ARMA models can be estimated. By ignoring the limitation on the frequencies of the forcing function, in the process of fitting a conventional ARMA model, one is liable derive estimates that are severely biased. The estimation biases can be avoided by sampling the continuous process at a rate corresponding to the maximum frequency of the forcing function. Then, there is a direct correspondence between the parameters of the bandlimited ARMA model and those of an equivalent continuoustime process.
FREQUENCYSELECTIVE FILTERS By D.S.G. POLLOCK
"... This paper shows how a frequencyselective filter that is applicable to short trended data sequences can be implemented via a frequencydomain approach. A filtered sequence can be obtained by multiplying the Fourier ordinates of the data by the ordinates of the frequency response of the filter and b ..."
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This paper shows how a frequencyselective filter that is applicable to short trended data sequences can be implemented via a frequencydomain approach. A filtered sequence can be obtained by multiplying the Fourier ordinates of the data by the ordinates of the frequency response of the filter and by applying the inverse Fourier transform to carry the product back into the time domain. Using this technique, it is possible, within the constraints of a finite sample, to design an ideal frequencyselective filter that will preserve all elements within a specified range of frequencies and that will remove all elements outside it. Approximations to ideal filters that are implemented in the time domain are commonly based on truncated versions of the infinite sequences of coefficients derived from the Fourier transforms of rectangular frequency response functions. An alternative to truncating an infinite sequence of coefficients is to wrap it around a circle of a circumference equal in length to the data sequence and to add the overlying coefficients. The coefficients of the wrapped filter can also be obtained by applying a discrete Fourier transform to a set of ordinates sampled from the frequency response function. Applying the coefficients to the data via circular convolution produces results that are identical to those obtained by a multiplication in the frequency domain, which constitutes a more efficient approach.
(2)
"... Whenever we form a linear combination of successive elements of a discretetime signal x(t) ={xt; t = ±1, ±2,...}, we are performing an operation that is described as linear filtering. In the case of a linear timeinvariant filter, such an operation can be represented by the equation (1) y(t) = ∑ ψj ..."
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Whenever we form a linear combination of successive elements of a discretetime signal x(t) ={xt; t = ±1, ±2,...}, we are performing an operation that is described as linear filtering. In the case of a linear timeinvariant filter, such an operation can be represented by the equation (1) y(t) = ∑ ψjx(t − j). j To assist in the algebraic manipulation of such equations, we may convert the infinite sequences x(t) and y(t) and the sequence of filter coefficients {ψj} into power series or polynomials. By associating zt to each element yt and by summing the sequence, we get
Large dimensional random k circulants
, 2009
"... Circulant matrices with general shift by k places have been studied in the literature and formula for their eigenvalues is known. We first reestablish this formula and some further properties of these eigenvalues in a manner suitable for our use. We then consider random k=k(n) circulants Ak,n with n ..."
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Circulant matrices with general shift by k places have been studied in the literature and formula for their eigenvalues is known. We first reestablish this formula and some further properties of these eigenvalues in a manner suitable for our use. We then consider random k=k(n) circulants Ak,n with n→ ∞ and whose input sequence{ai} is independent with mean zero and variance one and supn n−1 ∑ n i=1Eai  2+δ < ∞ for someδ>0. Under suitable restrictions on{k(n)}, we show that the limiting spectral distribution (LSD) of the empirical distribution of suitably scaled eigenvalues exists and identify the limits. As examples, (i) if kg =−1+ sn where g≥1 fixed and s=o(n 1/3), then the LSD is U1 ( ∏g
COVARIANCE ESTIMATION IN ELLIPTICAL MODELS WITH CONVEX STRUCTURE
"... We address structured covariance estimation in Elliptical distribution. We assume it is a priori known that the covariance belongs to a given convex set, e.g., the set of Toeplitz or banded matrices. We consider the General Method of Moments (GMM) optimization subject to these convex constraints. U ..."
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We address structured covariance estimation in Elliptical distribution. We assume it is a priori known that the covariance belongs to a given convex set, e.g., the set of Toeplitz or banded matrices. We consider the General Method of Moments (GMM) optimization subject to these convex constraints. Unfortunately, GMM is still nonconvex due to objective. Instead, we propose COCA a convex relaxation which can be efficiently solved. We prove that the relaxation is tight in the unconstrained case for a finite number of samples, and in the constrained case asymptotically. We then illustrate the advantages of COCA in synthetic simulations with structured Compound Gaussian distributions. In these examples, COCA outperforms competing methods as Tyler’s estimate and its projection onto a convex set. Index Terms — Elliptical distribution, Tyler’s scatter estimator, Generalized Method of Moments, robust covariance estimation.