In this article we solve a minimum problem involving step functions. The solution of this problem leads to an investigation into generalized h- and g-indices. This minimum problem and the related generalized h- and g-...
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In this article we solve a minimum problem involving step functions. The solution of this problem leads to an investigation into generalized h- and g-indices. This minimum problem and the related generalized h- and g-indices are studied in a general context of decreasing differentiable functions as well as in the specific case of Lotkaian informetrics. The study illustrates the use of h-and g-indices and their generalizations in a context which bears no relation to the research evaluation context in which these indices were originally introduced. (C) 2019 Elsevier Ltd. All rights reserved.
In this paper, we consider a step function characterized by a real-valued sequence and its linear expansion representation constructed via the matching pursuit (MP) algorithm. We utilize a waveform dictionary based on...
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In this paper, we consider a step function characterized by a real-valued sequence and its linear expansion representation constructed via the matching pursuit (MP) algorithm. We utilize a waveform dictionary based on the triangular function as part of this algorithm and representation. The waveform dictionary is comprised of waveforms localized in the time- frequency domain. In view of this, we prove that the triangular waveforms are more efficient than the rectangular waveforms used in a prior study by achieving a product of variances in the time-frequency domain closer to the lower bound of the Heisenberg Uncertainty Principle. We provide a MP algorithm solvable in polynomial time, contrasting the common exponential time when using Gaussian windows. We apply this algorithm on simulated data and real GDP data from 1947-2024 to demonstrate its application and efficiency.
An important task in analysing time series of the current passed by a single ion channel is the restoration of the true signal from the noisy recording. Hodgson [1] devised a reversible jump Markov chain Monte Carlo (...
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An important task in analysing time series of the current passed by a single ion channel is the restoration of the true signal from the noisy recording. Hodgson [1] devised a reversible jump Markov chain Monte Carlo (MCMC) algorithm to implement fully Bayesian single channel analysis. Noiseless ion channel signals are typically modelled as step functions with a discrete number of levels. As the number of steps is unknown, the joint posterior distribution under the Bayesian paradigm exhibits variable dimensionality, necessitating the use of reversible jump MCMC (Green [2]). This paper addresses the task of summarising posterior knowledge about the step function representing the noiseless channel signal, using Bayesian decision theory and the ideas of Rue [3]. We compute Bayes estimates of the true step function under a selection of loss functions from the same family.
Abstract: Let $f \in {L^1}(0,\infty ),\delta > 0$ and $({G_\delta }f)(t) = {\delta ^{ - 1}}\smallint _t^\infty {e^{(t - s)/\delta }}f(s)ds$. Given a partition $P = \{ 0 = {t_0} < {t_1} < \...
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Abstract: Let $f \in {L^1}(0,\infty ),\delta > 0$ and $({G_\delta }f)(t) = {\delta ^{ - 1}}\smallint _t^\infty {e^{(t - s)/\delta }}f(s)ds$. Given a partition $P = \{ 0 = {t_0} < {t_1} < \cdots < {t_i} < {t_{i + 1}} < \cdots \}$ of $[0,\infty )$ where ${t_i} \to \infty$, we approximate f by the step function ${A_P}f$ defined by \[ {A_P}f(t) = ({G_{{\delta _i}}}{G_{{\delta _{i - 1}}}} \cdots {G_{{\delta _i}}}f)(0)\quad {\text {for}}\;{t_{i - 1}} \leqslant t < {t_i},\] where ${\delta _i} = {t_i} - {t_{i - 1}}$. The main results concern several properties of this process, with the most important one being that ${A_P}f \to f$ in ${L^1}(0,\infty )$ as $\mu (P) = {\sup _i}{\delta _i} \to 0$. An application to difference approximations of evolution problems is sketched.
The author introduces a new class of step functions and defines them in terms of the Rademacher functions. Two complete systems of orthogonal slant step functions are constructed and their characteristics are analyzed...
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The author introduces a new class of step functions and defines them in terms of the Rademacher functions. Two complete systems of orthogonal slant step functions are constructed and their characteristics are analyzed. The relationship between the basis functions of these systems and the system of the Walsh functions is established and their orthonormality is proved. It is shown that the proposed systems of slant step functions can be efficiently used to code video signal.
The methods presented in this paper provide an improved technique for fitting a continuous function to discontinuous step functions so often evolving from empirical studies such as estimation of demand and supply func...
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The methods presented in this paper provide an improved technique for fitting a continuous function to discontinuous step functions so often evolving from empirical studies such as estimation of demand and supply functions by linear programming. Curve fitting to step functions is analyzed using the criterion of minimizaton of the integral of squared distances between the curve being fitted and a step function. The criterion is a limiting form of least squares fit where all points on the step function are treated as “observations.” Derivations are given for equations specifying the minimum integral fit for polynomials, a criterion of fit analogous to the coefficient of multiple determination in least squares fitting, and the analogue of the standard error of estimate in least squares.
New methods of approximation of step functions with an estimation of the error of the approximation are suggested. The suggested methods do not have any of the disadvantages of traditional approximations of step funct...
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The parameter estimation to mixture models has been shown as a local optimal solution for decades. In this paper, we propose a functional estimation to mixture models using step functions. We show that the proposed fu...
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ISBN:
(纸本)9781467395038
The parameter estimation to mixture models has been shown as a local optimal solution for decades. In this paper, we propose a functional estimation to mixture models using step functions. We show that the proposed functional inference yields a convex formulation and consequently the mixture models are feasible for a global optimum inference. The proposed approach further unifies the existing isolated exemplar-based clustering techniques at a higher level of generality, e.g. it provides a theoretical justification for the heuristics of the clustering by affinity propagation Frey & Dueck (2007);it reproduces Lashkari & Golland (2007)'s convex formulation as a special case under this step function construction. Empirical studies also verify the theoretic justifications.
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