Laguerre Expansion Research Articles

Integral pulse frequency modulation model driven by sympathovagal dynamics: Synthetic vs. real heart rate variability

Computational models that generate synthetic heart rate variability (HRV) series constitute important tools for the assessment of the effect of autonomic nervous system activity on cardiovascular control, and for the evaluation of novel algorithms using synthetic data. A widely used technique for synthetic HRV generation is the integral pulse frequency modulation (IPFM) model; however, IPFM relies on the HRV spectral paradigm, which cannot separate sympathetic and vagal oscillations that are overlapped in the low-frequency band (0.04–0.15 Hz). To overcome this limitation, a novel IPFM-inspired model driven by cardiac sympathetic and vagal dynamics estimated from HRV is proposed, where our recently developed sympathetic and parasympathetic activity indices that rely on orthonormal Laguerre expansions of the RR interval autoregressive kernels are exploited. The performance of the proposed model is evaluated by comparing the synthetic vs. real RR interval series in a simulation study involving postural changes, with real HRV data gathered from 10 healthy subjects. Moreover, the performance of the proposed model is compared with that of the standard IPFM to discern different autonomic control states associated with resting and postural changes. The results confirm that the proposed physiologically inspired model adequately predicts RR intervals during resting and postural changes. The proposed model clearly outperforms the standard IPFM method, considering both median error, and maximum error. The developed model provides valuable insights for a better understanding of the sympathovagal activity in the analysis of heartbeat dynamics.

On Synthetic Instrument Response Functions of Time-Correlated Single-Photon Counting Based Fluorescence Lifetime Imaging Analysis

Time-correlated single-photon counting (TCSPC) has been the gold standard for fluorescence lifetime imaging (FLIM) techniques due to its high signal-to-noize ratio and high temporal resolution. The sensor system's temporal instrument response function (IRF) should be considered in the deconvolution procedure to extract the real fluorescence decay to compensate for the distortion on measured decays contributed by the system imperfections. However, to measure the instrument response function is not trivial, and the measurement setup is different from measuring the real fluorescence. On the other hand, automatic synthetic IRFs can be directly derived from the recorded decay profiles and provide appropriate accuracy. This paper proposed and examined a synthetic IRF strategy. Compared with traditional automatic synthetic IRFs, the new proposed automatic synthetic IRF shows a broader dynamic range and better accuracy. To evaluate its performance, we examined simulated data using nonlinear least square deconvolution based on both the Levenberg-Marquardt algorithm and the Laguerre expansion method for bi-exponential fluorescence decays. Furthermore, experimental FLIM data of cells were also analyzed using the proposed synthetic IRF. The results from both the simulated data and experimental FLIM data show that the proposed synthetic IRF has a better performance compared to traditional synthetic IRFs. Our work provides a faster and precise method to obtain IRF, which may find various FLIM-based applications. We also reported in which conditions a measured or a synthesized IRF can be applied.

Estimation of multivariate generalized gamma convolutions through Laguerre expansions.

The generalized gamma convolution class of distribution appeared in Thorin's work while looking for the infinite divisibility of the log-Normal and Pareto distributions. Although these distributions have been extensively studied in the univariate case, the multivariate case and the dependence structures that can arise from it have received little interest in the literature. Furthermore, only one projection procedure for the univariate case was recently constructed, and no estimation procedure are available. By expending the densities of multivariate generalized gamma convolutions into a tensorized Laguerre basis, we bridge the gap and provide performant estimations procedures for both the univariate and multivariate cases. We provide some insights about performance of these procedures, and a convergent series for the density of multivariate gamma convolutions, which is shown to be more stable than Moschopoulos's and Mathai's univariate series. We furthermore discuss some examples.

Statistical Feature Extraction and System Identification Algorithms for Partial Discharge Signal Classification Using Laguerre Polynomial Expansion

In this paper, a novel algorithm for partial discharge (PD) pulse waveform analysis and separation based on orthogonal series expansion using Laguerre polynomials is proposed. A system identification technique is also developed to help increase the accuracy rate of the signal classification. Laguerre polynomial series expansions are applied on the PD pulses to effectively extract important information and well-discriminative features. Laguerre expansion coefficients inherently contain information about the amplitude (intensity) and the relaxation time of a dynamic system and this helps direct characterization of PD pulses. The proposed technique is applied to PD pulses acquired from a laboratory set of multiple PD sources. It is shown that this statistical method can classify partial discharge signals with high accuracies, even when the signals are visually indistinguishable. In this research, two methods are developed for system identification. One is based on a deterministic approach in the form of a recursive formula and the other one is a stochastic method based on a group Lasso methodology. The aim of system identification is to improve the classification accuracy by removing the effect of the measurement system from the observed PD waveforms. The accuracy rate of the proposed algorithm is evaluated using the signals that are captured from simultaneous artificial defects in high pressure SF <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">6</inf> and in low pressure air.

Laguerre Expansion Series Based Reduced Order Interval Systems

Model order reduction techniques have attracted researchers for the analysis of higher-order systems. The brief presents the order reduction of the interval system in state-space form using the Laguerre polynomial method with singular value decomposition. Laguerre - SVD model reduction technique is approved for the algorithms reduced computational complexity and stability perseverance. The proposed algorithm applies Laguerre approximation to interval system to compute an interval Krylov matrix using the Interval laboratory MATLAB toolbox. Then singular value decomposition is performed on the concatenated matrix with the upper limit and lower limit of the obtained interval Krylov matrix. The resultant column orthogonal matrix yields the reduced interval system through congruence transformation. The technique preserves the original system properties and is proved to be stable. Pertinent numerical results validate the proposed technique with examples from literature.

A gyrokinetic model for the plasma periphery of tokamak devices

A gyrokinetic model is presented that can properly describe large and small amplitude electromagnetic fluctuations occurring on scale lengths ranging from the electron Larmor radius to the equilibrium perpendicular pressure gradient scale length, and the arbitrarily large deviations from thermal equilibrium that are present in the plasma periphery of tokamak devices. The formulation of the gyrokinetic model is based on a second-order accurate description of the single charged particle dynamics, derived from Lie perturbation theory, where the fast particle gyromotion is decoupled from the slow drifts assuming that the ratio of the ion sound Larmor radius to the perpendicular equilibrium pressure scale length is small. The collective behaviour of the plasma is obtained by a gyrokinetic Boltzmann equation that describes the evolution of the gyroaveraged distribution function. The collisional effects are included by a nonlinear gyrokinetic Dougherty collision operator. The gyrokinetic model is then developed into a set of coupled fluid equations referred to as the gyrokinetic moment hierarchy. To obtain this hierarchy, the gyroaveraged distribution function is expanded onto a Hermite–Laguerre velocity-space polynomial basis. Then, the gyrokinetic equation is projected onto the same basis yielding the spatial and temporal evolution of the Hermite–Laguerre expansion coefficients. A closed set of fluid equations for the lowest-order coefficients is presented. The Hermite–Laguerre projection is performed accurately at arbitrary perpendicular wavenumber values. Finally, the self-consistent evolution of the electromagnetic fields is described by a set of gyrokinetic Maxwell equations derived from a variational principle where the velocity integrals are explicitly evaluated.

On the Padé and Laguerre–Tricomi–Weeks Moments Based Approximations of the Scale Function W and of the Optimal Dividends Barrier for Spectrally Negative Lévy Risk Processes

This paper considers the Brownian perturbed Cramér–Lundberg risk model with a dividends barrier. We study various types of Padé approximations and Laguerre expansions to compute or approximate the scale function that is necessary to optimize the dividends barrier. We experiment also with a heavy-tailed claim distribution for which we apply the so-called “shifted” Padé approximation.

Nonlinear gyrokinetic Coulomb collision operator

A gyrokinetic Coulomb collision operator is derived, which is particularly useful to describe the plasma dynamics at the periphery region of magnetic confinement fusion devices. The derived operator is able to describe collisions occurring in distribution functions arbitrarily far from equilibrium with variations on spatial scales at and below the particle Larmor radius. A multipole expansion of the Rosenbluth potentials is used in order to derive the dependence of the full Coulomb collision operator on the particle gyroangle. The full Coulomb collision operator is then expressed in gyrocentre phase-space coordinates, and a closed formula for its gyroaverage in terms of the moments of the gyrocentre distribution function in a form ready to be numerically implemented is provided. Furthermore, the collision operator is projected onto a Hermite–Laguerre velocity space polynomial basis and expansions in the small electron-to-ion mass ratio are provided.

Numerical Study for System of ODEs Obtained from MHD Flow Past a Permeable Flat Plate in a Darcian Porous Medium Using Laguerre Collocation Method

In this article, we introduce a numerical solution for the system of ODEs which governs the problem of magnetohydrodynamic boundary layer flow and heat transfer past a permeable flat plate which embedded in a porous medium by using an efficient Laguerre collocation method. The controlling parameters which affect the dimensionless velocity are the Darcy number, the Reynolds number and the magnetic number, whereas, the radiation parameter and Peclet number are the emerging parameters on the dimensionless temperature. The influence of these parameters on both the dimensionless velocity and the dimensionless temperature are analyzed graphically. The proposed method is based on replacement of the unknown function by a truncated series of the well-known Laguerre expansion of functions. An approximate formula of the integer derivative is introduced. The introduced method converts the proposed equation by means of collocation points to a system of algebraic equations with Laguerre coefficients. Verification of the numerical Laguerre collocation method is tested after introducing the comparison with the previously published results.

On the connection coefficients and recurrence relations arising from expansions in series of modified generalized Laguerre polynomials: Applications on a semi-infinite domain

Abstract Herein, three important theorems were stated and proved. The first relates the modified generalized Laguerre expansion coefficients of the derivatives of a function in terms of its original expansion coefficients; and an explicit expression for the derivatives of modified generalized Laguerre polynomials of any degree and for any order as a linear combination of modified generalized Laguerre polynomials themselves is also deduced. The second theorem gives new modified generalized Laguerre coefficients of the moments of one single modified generalized Laguerre polynomials of any degree. Finally, the third theorem expresses explicitly the modified generalized Laguerre coefficients of the moments of a general-order derivative of an infinitely differentiable function in terms of its modified generalized Laguerre coefficients. Some spectral applications of these theorems for solving ordinary differential equations with varying coefficients and some specific applied differential problems, by reducing them to recurrence relations in their expansion coefficients of the solution are considered.

A non-structural investigation of VIX risk neutral density

Abstract We propose a non-structural method to retrieve the risk-neutral density (RND) implied by options on the CBOE Volatility Index (VIX). The methodology is based on orthogonal polynomial expansions around a kernel density and yields the RND of the underlying asset without the need for a parametric specification. The classic family of Laguerre expansions is extended to include the GIG and the generalized Weibull kernels. We show that orthogonal polynomial expansions yield accurate approximations of the RND of VIX and they generally outperform commonly used non-parametric methods when controlling for accuracy. Based on a panel of observed VIX options, we retrieve the variance swap term structure, the time series of VVIX, the VIX risk-neutral moments and the Volatility-at-Risk, which reveal a number of stylized facts on the RND of VIX.

Hardy\u2019s Inequality for Laguerre Expansions of Hermite Type

Hardy’s inequality for Laguerre expansions of Hermite type with the index alpha in ({-1/2}cup [1/2,infty ))^d is proved in the multi-dimensional setting with the exponent 3d / 4. We also obtain the sharp analogue of Hardy’s inequality with L^1 norm replacing H^1 norm at the expense of increasing the exponent by an arbitrarily small value.

Quasi-Optimal Rank-Structured Approximation to Multidimensional Parabolic Problems by Cayley Transform and Chebyshev Interpolation

Abstract In the present paper we propose and analyze a class of tensor approaches for the efficient numerical solution of a first order differential equation ψ ′ ⁢ ( t ) + A ⁢ ψ = f ⁢ ( t ) {\psi^{\prime}(t)+A\psi=f(t)} with an unbounded operator coefficient A. These techniques are based on a Laguerre polynomial expansions with coefficients which are powers of the Cayley transform of the operator A. The Cayley transform under consideration is a useful tool to arrive at the following aims: (1) to separate time and spatial variables, (2) to switch from the continuous “time variable” to “the discrete time variable” and from the study of functions of an unbounded operator to the ones of a bounded operator, (3) to obtain exponentially accurate approximations. In the earlier papers of the authors some approximations on the basis of the Cayley transform and the N-term Laguerre expansions of the accuracy order 𝒪 ⁢ ( e - N ) {\mathcal{O}(e^{-N})} were proposed and justified provided that the initial value is analytical for A. In the present paper we combine the Cayley transform and the Chebyshev–Gauss–Lobatto interpolation and arrive at an approximation of the accuracy order 𝒪 ⁢ ( e - N ) {\mathcal{O}(e^{-N})} without restrictions on the input data. The use of the Laguerre expansion or the Chebyshev–Gauss–Lobatto interpolation allows to separate the time and space variables. The separation of the multidimensional spatial variable can be achieved by the use of low-rank approximation to the Cayley transform of the Laplace-like operator that is spectrally close to A. As a result a quasi-optimal numerical algorithm can be designed.

Vector-valued extensions for fractional integrals of Laguerre expansions

We prove some vector-valued inequalities for fractional integrals defined for several orthonormal systems of Laguerre functions. On the one hand, we obtain weighted Lp-Lq vector-valued extensions, in a multidimensional setting, for negative powers of the operator related to so-called Laguerre expansions of Hermite type. On the other hand, we give necessary and sufficient conditions for vector-valued Lp-Lq estimates related to negative powers of the Laguerre operator associated to expansions of convolution type, in a one-dimensional setting. Both types of vector-valued inequalities are based on estimates of the kernel with precise control of the parameters involved. As an application, mixed norm estimates for fractional integrals related to the harmonic oscillator are deduced. © Instytut Matematyczny PAN, 2018.

Orthogonal expansions for VIX options under affine jump diffusions

In this work we derive new closed-form pricing formulas for VIX options in the jump-diffusion SVJJ model proposed by Duffie et al. [Econometrica, 2000, 68, 1343–1376]. Our approach is based on the classic methodology of approximating a density function with an orthogonal expansion of polynomials weighted by a kernel. Orthogonal expansions based on the Gaussian distribution, such as Edgeworth or Gram–Charlier expansions, have been successfully employed by a number of authors in the context of equity options. However, these expansions are not quite suitable for volatility or variance densities as they inherently assign positive mass to the negative real line. Here we approximate option prices via expansions that instead are based on kernels defined on the positive real line. Specifically, we consider a flexible family of distributions, which generalizes the gamma kernel associated with the classic Laguerre expansions. The method can be employed whenever the moments of the underlying variance distribution are known. It provides fast and accurate price computations, and therefore it represents a valid and possibly more robust alternative to pricing techniques based on Fourier transform inversions.

Besov and Triebel–Lizorkin spaces associated with Laguerre expansions of Hermite type

Homogeneous Besov and Triebel-Lizorkin spaces associated with multi-dimensional Laguerre function expansions of Hermite type with index $\alpha \in [-1/2,\infty)^d\backslash (-1/2,1/2)^d$, $d\geq 1$, are defined and investigated. To achieve expected goals Schwartz type spaces on $R^d_+$ are introduced and then tempered type distributions are constructed. Also, ideas from a recent paper of Bui and Duong on Besov and Triebel-Lizorkin spaces associated with Hermite functions expansions are used. This means, in particular, using molecular decomposition and an appropriate form of a Calder\'on reproducing formula.

A new multi-domain method based on an analytical control surface for linear and second-order mean drift wave loads on floating bodies

A novel multi-domain method based on an analytical control surface is proposed by combining the use of free-surface Green function and Rankine source function. A cylindrical control surface is introduced to subdivide the fluid domain into external and internal domains. Unlike the traditional domain decomposition strategy or multi-block method, the control surface here is not panelized, on which the velocity potential and normal velocity components are analytically expressed as a series of base functions composed of Laguerre function in vertical coordinate and Fourier series in the circumference. Free-surface Green function is applied in the external domain, and the boundary integral equation is constructed on the control surface in the sense of Galerkin collocation via integrating test functions orthogonal to base functions over the control surface. The external solution gives rise to the so-called Dirichlet-to-Neumann[DN2] and Neumann-to-Dirichlet[ND2] relations on the control surface. Irregular frequencies, which are only dependent on the radius of the control surface, are present in the external solution, and they are removed by extending the boundary integral equation to the interior free surface (circular disc) on which the null normal derivative of potential is imposed, and the dipole distribution is expressed as Fourier–Bessel expansion on the disc. In the internal domain, where the Rankine source function is adopted, new boundary integral equations are formulated. The point collocation is imposed over the body surface and free surface, while the collocation of the Galerkin type is applied on the control surface. The present method is valid in the computation of both linear and second-order mean drift wave loads. Furthermore, the second-order mean drift force based on the middle-field formulation can be calculated analytically by using the coefficients of the Fourier–Laguerre expansion.

On the correspondence of hyperbolic geometry and system analysis

Different aspects of the relation between hyperbolic geometry and linear system theory are discussed in this paper. The underlying connection is presented by an intuitive example that points out the basic motivations. It is shown that the convergence factor of Laguerre series expansion is equal to the hyperbolic distance, under certain conditions. Preliminary results are also reported, connecting the H∞ norm and ν-gap metric with the hyperbolic distance. Furthermore, the equivalence of (i) the H∞ norm of the difference of two first order LTI system, (ii) the ν-gap of these systems and (iii) the hyperbolic distance is also proved, under specified assumptions.

Volatility-Related Exchange Traded Assets: An Econometric Investigation

We develop a theoretical framework for covariance stationary but persistent positively valued processes which combines a semi-nonparametric expansion of the Gamma distribution with a component version of the multiplicative error model. Our conditional mean assumption allows for slow, possibly nonmonotonic mean-reversion, while our distribution assumption provides more flexibility than a traditional Laguerre expansion while preserving positivity of the density. We apply our framework to a dynamic portfolio allocation for Exchange Traded Notes tracking short- and mid-term VIX futures indices, which are increasingly popular but risky financial instruments. We show the superior performance of the strategies based on our econometric model.

Generalized functions and Laguerre expansions

In this paper we study and characterize expansions of distributions in the Zemanian spaces, \({\mathcal {H}}_{\mu }\) and its dual \({\mathcal {H}}_{\mu }^{\prime }\) (\(\mu \ge -\frac{1}{2}\)) with respect to Laguerre functions. We obtain as applications of this result, the kernel Theorem and a structure Theorem for \({\mathcal {H}}_{\mu }^{\prime }\). We also introduce a new algebra of generalized functions in the sense of J. F. Colombeau such that it satisfies interesting properties involving the Hankel transformation and Hankel convolution.