Gaussian Series Research Articles

Correlation functions for zeros of a Gaussian power series and Pfaffians

We show that the zeros of the random power series with i.i.d. real Gaussian coefficients form a Pfaffian point process. We further show that the product moments for absolute values and signatures of the power series can also be expressed by Pfaffians.

A New Algorithm for Blind Signal Separation Based on Relative Newton Method and Maximum Likelihood Criterion

A new algorithm of blind signal separation that jointly exploits the selection of rational nonlinear functions and relative Newton method is proposed. This algorithm uses rational nonlinear functions in constructing the cost function, which have less computational complexity than the usual nonlinear functions. We use relative Newton method to solve the solution procedure of the cost function based on maximum likelihood criterion. The source data for simulation are taken from generalized Gaussian distribution series, as well as realistic voice signal. Simulation results show superior performance of the proposed algorithm compared with classical ones such as JadeR and fastica.

Rates of contraction for posterior distributions in Lr-metrics, 1 ≤ r ≤ ∞

The frequentist behavior of nonparametric Bayes estimates, more specifically, rates of contraction of the posterior distributions to shrinking $L^r$-norm neighborhoods, $1\le r\le\infty$, of the unknown parameter, are studied. A theorem for nonparametric density estimation is proved under general approximation-theoretic assumptions on the prior. The result is applied to a variety of common examples, including Gaussian process, wavelet series, normal mixture and histogram priors. The rates of contraction are minimax-optimal for $1\le r\le2$, but deteriorate as $r$ increases beyond 2. In the case of Gaussian nonparametric regression a Gaussian prior is devised for which the posterior contracts at the optimal rate in all $L^r$-norms, $1\le r\le\infty$.

Synthesis of multivariate stationary series with prescribed marginal distributions and covariance using circulant matrix embedding

The problem of synthesizing multivariate stationary series Y[ n]=( Y 1[ n],…, Y P [ n]) T , n ∈ Z , with prescribed non-Gaussian marginal distributions, and a targeted covariance structure, is addressed. The focus is on constructions based on a memoryless transformation Y p [ n]= f p ( X p [ n]) of a multivariate stationary Gaussian series X[ n]=( X 1[ n],…, X P [ n]) T . The mapping between the targeted covariance and that of the Gaussian series is expressed via Hermite expansions. The various choices of the transforms f p for a prescribed marginal distribution are discussed in a comprehensive manner. The interplay between the targeted marginal distributions, the choice of the transforms f p , and on the resulting reachability of the targeted covariance, is discussed theoretically and illustrated on examples. Also, an original practical procedure warranting positive definiteness for the transformed covariance at the price of approximating the targeted covariance is proposed, based on a simple and natural modification of the popular circulant matrix embedding technique. The applications of the proposed methodology are also discussed in the context of network traffic modeling. Matlab codes implementing the proposed synthesis procedure are publicly available at http://www.hermir.org.

Gibbs measure for the periodic derivative nonlinear Schrödinger equation

In this paper we construct a Gibbs measure for the derivative Schrödinger equation on the circle. The construction uses some renormalizations of Gaussian series and Wiener chaos estimates, ideas which have already been used by the second author in a work on the Benjamin–Ono equation.

Adaptive wavelet decompositions of stationary time series

A general and flexible framework for the wavelet-based decompositions of stationary time series in discrete time, called adaptive wavelet decompositions (AWDs), is introduced. It is shown that several particular AWDs can be constructed with the aim of providing decomposition (approximation and detail) coefficients that exhibit certain nice statistical properties, where the latter can be chosen based on a range of theoretical or applied considerations. AWDs make use of a Fast Wavelet Transform-like algorithm whose filters - in contrast with their counterparts in Orthogonal Wavelet Decompositions (OWDs) – may depend on the scale. As with OWDs, this algorithm has good properties such as computational efficiency and invariance to polynomial trends. A property whose pursuit plays a central role in this work is the decorrelation of the detail coefficients. For many time series models (e.g., FARIMA(0,δ,0)), the AWD filters can be defined so that the resulting AWD detail coefficients are all (exactly) decorrelated. The corresponding AWDs, called Exact AWDs (EAWDs), are particularly useful in simulation of Gaussian stationary time series, if the associated filters have a fast decay. The proposed simulation methods generalize and improve upon existing wavelet-based ones. AWDs for which the detail coefficients are not exactly decorrelated, but still more decorrelated than those of OWDs, are referred to as approximate AWDs (AAWDs). They can be obtained by truncating EAWD filters, or by adopting some of the existing approaches to modeling the dependence structure of the OWD detail coefficients (e.g., Craigmile et al., 2005). AAWDs naturally lead to new wavelet-based Maximum Likelihood estimators. The performance of these estimators is investigated through simulations and from some theoretical standpoints. The focus in estimation is also on Gaussian stationary series, though the method is expected to work for non-Gaussian stationary series as well.

Comparison between different Gaussian series representations of the imaginary time propagator

A useful approximation for the thermal operator exp(-betaH[over ]) is based on its representation in terms of either frozen or thawed Gaussian states. Such approximate representations are leading-order terms in respective series representations of the thermal operator. A numerical study of the convergence properties of the frozen Gaussian series representation has been recently published. In this paper, we extend the previous study to include also the convergence properties of the more expensive thawed Gaussian series representation of the thermal operator. We consider three different formulations for the series representation and apply them to a quartic double-well potential to find that the thawed Gaussian series representation converges faster than the frozen Gaussian one. Further analysis is presented as to the convergence properties and the numerical efficiency of three different thawed Gaussian series representation. The unsymmetrized form converges most rapidly, however, the lower order approximations of the symmetrized forms are more accurate. Comparison with a standard discretized path-integral evaluation demonstrates that the Gaussian based perturbation series representation converges much faster.

A Novel Analysis of Microstrip Structures Using the Gaussian Green's Function Method

A novel closed form expression is derived for spatial Green's functions of microstrip structures by expanding the spectral Green's function into a Gaussian series. This innovative method is called the Gaussian Green's function (GGF) method due to the Gaussian form in the closed form Green's function representation. The main advantage of the GGF method lies in its precision as well as rapid convergence. To demonstrate the versatility of this method, the current distribution of a microstrip antenna is achieved via the combination of the method of moments (MoM) and GGF method. It is shown that this method can be computationally very efficient with less than 1% error compared to the numerical integration of the spectral integral. Also, the results of the GGF method have been compared to the results of the commercial full-wave software of Agilent ADS.

The uselessness of the Fast Gauss Transform for summing Gaussian radial basis function series

The Fast Gauss Transform is an algorithm for summing a series of Gaussians which is sometimes much faster than direct summation. Gaussian series in d dimensions are of the form ∑ j = 1 N λ j exp ( - [ α / h ] 2 ‖ x - x j ‖ 2 ) where the x j are the centers, h is the average separation between centers and α is the relative inverse width parameter. We show that the speed-up of the Fast Gauss Transform is bounded by a factor Ω ( α ) . When α ≪ 1 , Ω can be large. However, when applied to Gaussian radial basis function interpolation, it is difficult to apply the Gaussian basis in this parameter range because the interpolation matrix is exponentially ill-conditioned: the condition number κ ∼ ( 1 / 2 ) exp π 2 4 α 2 for a uniform, one-dimensional grid, and larger still in two dimensions or when the grid is irregular. Furthermore, the Gaussian RBF interpolant is ill-conditioned for most series in the sense that the interpolant is the small difference of terms with exponentially large coefficients. Fornberg and Piret developed a “QR-basis” that ameliorates this difficulty for approximations on the surface of a sphere, but because the recombined basis functions are perturbed spherical harmonics, not Gaussians, the Fast Gauss Transform is no longer applicable. The solution of the interpolation matrix system by a preconditioned iteration is less sensitive to condition numbers than direct methods because iterations are self-correcting and also because the preconditioning reduces the spread of the eigenvalues. However, each iteration requires a matrix–vector multiply which is fast only if this operation can be performed by some species of Fast Summation. When α ∼ O ( 1 ) , alas, we show that Ω is not large and the Fast Gauss Transform is not accelerative. Gaussian RBFs are unusual among RBF species through the absence of long-range interactions due to the exponential decay of the Gaussians with distance from their centers; many other RBF species do have long-range interactions, and it is well-established that these can be accelerated by fast multipole and treecode algorithms. We offer a less rigorous scale analysis argument to explain why the underlying difficulty in accelerating short-range interactions is not peculiar to the Gaussian RBF basis or to the Fast Gauss Transform, but rather is likely to be a generic difficulty in accelerating the short-range interactions of almost any RBF basis with almost any Fast Summation.

Frozen Gaussian series representation of the imaginary time propagator theory and numerical tests

Thawed Gaussian wavepackets have been used in recent years to compute approximations to the thermal density matrix. From a numerical point of view, it is cheaper to employ frozen Gaussian wavepackets. In this paper, we provide the formalism for the computation of thermal densities using frozen Gaussian wavepackets. We show that the exact density may be given in terms of a series, in which the zeroth order term is the frozen Gaussian. A numerical test of the methodology is presented for deep tunneling in the quartic double well potential. In all cases, the series is observed to converge. The convergence of the diagonal density matrix element is much faster than that of the antidiagonal one, suggesting that the methodology should be especially useful for the computation of partition functions. As a by product of this study, we find that the density matrix in configuration space can have more than two saddle points at low temperatures. This has implications for the use of the quantum instanton theory.

A copula-based Markov chain model for the analysis of binary longitudinal data

A fully parametric first-order autoregressive (AR(1)) model is proposed to analyse binary longitudinal data. By using a discretized version of a copula, the modelling approach allows one to construct separate models for the marginal response and for the dependence between adjacent responses. In particular, the transition model that is focused on discretizes the Gaussian copula in such a way that the marginal is a Bernoulli distribution. A probit link is used to take into account concomitant information in the behaviour of the underlying marginal distribution. Fixed and time-varying covariates can be included in the model. The method is simple and is a natural extension of the AR(1) model for Gaussian series. Since the approach put forward is likelihood-based, it allows interpretations and inferences to be made that are not possible with semi-parametric approaches such as those based on generalized estimating equations. Data from a study designed to reduce the exposure of children to the sun are used to illustrate the methods.

Fatigue damage assessment for a spectral model of non-Gaussian random loads

In this paper, a new model for random loads–the Laplace driven moving average–is presented. The model is second order, non-Gaussian, and strictly stationary. It shares with its Gaussian counterpart the ability to model any spectrum but has additional flexibility to model the skewness and kurtosis of the marginal distribution. Unlike most other non-Gaussian models proposed in the literature, such as the transformed Gaussian or Volterra series models, the new model is no longer derivable from Gaussian processes. In the paper, a summary of the properties of the new model is given and its upcrossing intensities are evaluated. Then it is used to estimate fatigue damage both from simulations and in terms of an upper bound that is of particular use for narrowband spectra.

Calculating the autocovariances and the likelihood for periodic V ARMA models

This paper concerns the autocovariance calculation and likelihood evaluation for periodic vector ARMA models (PV ARMA). Based on a state space representation of PV ARMA models, we derive an algorithm for computing the PV ARMA autocovariances. The proposed method computes the autocovariances for distinct seasons separately, thereby facilitating efficient calculation for models with a large period. As a result, the obtained autocovariance calculation procedure is exploited in a periodic Chandrasekhar-type filter to evaluate the exact likelihood for Gaussian PV ARMA series. Empirical evidence shows the superiority of the periodic Chandrasekhar algorithm for likelihood evaluation over the Kalman-based one.

Significance tests for the wavelet cross spectrum and wavelet linear coherence

Abstract. This work attempts to develop significance tests for the wavelet cross spectrum and the wavelet linear coherence as a follow-up study on Ge (2007). Conventional approaches that are used by Torrence and Compo (1998) based on stationary background noise time series were used here in estimating the sampling distributions of the wavelet cross spectrum and the wavelet linear coherence. The sampling distributions are then used for establishing significance levels for these two wavelet-based quantities. In addition to these two wavelet quantities, properties of the phase angle of the wavelet cross spectrum of, or the phase difference between, two Gaussian white noise series are discussed. It is found that the tangent of the principal part of the phase angle approximately has a standard Cauchy distribution and the phase angle is uniformly distributed, which makes it impossible to establish significance levels for the phase angle. The simulated signals clearly show that, when there is no linear relation between the two analysed signals, the phase angle disperses into the entire range of [−π,π] with fairly high probabilities for values close to ±π to occur. Conversely, when linear relations are present, the phase angle of the wavelet cross spectrum settles around an associated value with considerably reduced fluctuations. When two signals are linearly coupled, their wavelet linear coherence will attain values close to one. The significance test of the wavelet linear coherence can therefore be used to complement the inspection of the phase angle of the wavelet cross spectrum. The developed significance tests are also applied to actual data sets, simultaneously recorded wind speed and wave elevation series measured from a NOAA buoy on Lake Michigan. Significance levels of the wavelet cross spectrum and the wavelet linear coherence between the winds and the waves reasonably separated meaningful peaks from those generated by randomness in the data set. As with simulated signals, nearly constant phase angles of the wavelet cross spectrum are found to coincide with large values in the wavelet linear coherence between the winds and the waves. Not limited to geophysics, the significance tests developed in the present work can also be applied to many other quantitative studies using the continuous wavelet transform.

Numerical Approximation of Some Infinite Gaussian Series and Integrals

The paper deals with numerical computation of the asymptotic variance of the so-called increment ratio (IR) statistic and its modifications. The IR statistic is useful for estimation and hypothesis testing on fractional parameter H ∈ (0, 1) of random process (time series), see Surgailis et al. [1], Bardet and Surgailis [2]. The asymptotic variance of the IR statistic is given by an infinite integral (or infinite series) of 4-dimensional Gaussian integrals which depend on parameter H. Our method can be useful for numerical computation of other similar slowly convergent Gaussian integrals/series. Graphs and tables of approximate values of the variances σp2(H) and σˆp2(H), p = 1, 2 are included.

Gaussian approximation of exponential type orbitals based on B functions

AbstractThis work gives new, highly accurate optimized gaussian series expansions for the B functions used in molecular quantum mechanics. These functions are generally chosen because of their compact Fourier transform, following Shavitt. The inverse Laplace transform in the square root of the variable is used for Gauss quadrature in this work. Two procedures for obtaining accurate gaussian expansions have been compared for the required extended precision arithmetic. The first is based on Gaussian quadratures and the second on direct optimization. Both use the Maple computer algebra system. Numerical results are tabulated and compared with previous work. Special cases are found to agree before pushing the optimization technique further. The optimal gaussian expansions of B functions obtained in this work are available for reference. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009

$L^p$ properties for Gaussian random series

Let c = (c n ) n ∈N* be an arbitrary sequence of l 2 (N*) and let F c (ω) be a random series of the type F c (ω) =Σn∈N* g n (ω)c n e n , where (g n ) n ∈N* is a sequence of independent N c (0,1) Gaussian random variables and (e n ) n ∈N* an orthonormal basis of L 2 (Y,Μ,μ) (the finite measure space (Y,M,μ) being arbitrary). By using the equivalence of Gaussian moments and an integrability theorem due to Fernique, we show that a necessary and sufficient condition for F c (ω) to belong to L p (Y, M,μ), p ∈ [2, oo) for any c ∈ l 2 (N*) almost surely is that sup n∈ N* ∥e n ∥L p (Y,M,μ) < oo. One of the main motivations behind this result is the construction of a nontrivial Gibbs measure invariant under the flow of the cubic defocusing nonlinear Schrodinger equation posed on the open unit disc of R 2 .

Best Approximation of Gaussian Neural Networks With Nodes Uniformly Spaced

This paper is aimed at exposing the reader to certain aspects in the design of the best approximants with Gaussian radial basis functions (RBFs). The class of functions to which this approach applies consists of those compactly supported in frequency. The approximative properties of uniqueness and existence are restricted to this class. Functions which are smooth enough can be expanded in Gaussian series converging uniformly to the objective function. The uniqueness of these series is demonstrated by the context of the orthonormal basis in a Hilbert space. Furthermore, the best approximation to a given band-limited function from a truncated Gaussian series is analyzed by an energy-based argument. This analysis not only gives a theoretical proof concerned with the existence of best approximations but addresses the problems of architectural selection. Specifically, guidance for selecting the variance and the oversampling parameters is provided for practitioners.

Establishing the stochastic nature of intracellular calcium oscillations from experimental data

Calcium has been established as a key messenger in both intra- and intercellular signaling. Experimentally observed intracellular calcium responses to different agonists show a variety of behaviors from simple spiking to complex oscillatory regimes. Here we study typical experimental traces of calcium oscillations in hepatocytes obtained in response to phenylephrine and ATP. The traces were analyzed with methods of nonlinear time series analysis in order to determine the stochastic/deterministic nature of the intracellular calcium oscillations. Despite the fact that the oscillations appear, visually, to be deterministic yet perturbed by noise, our analyses provide strong evidence that the measured calcium traces in hepatocytes are prevalently of stochastic nature. In particular, bursting calcium oscillations are temporally correlated Gaussian series distorted by a monotonic, instantaneous, time-independent function, whilst the spiking behavior appears to have a dynamical nonlinear component whereby the overall determinism level is still low. The biological importance of this finding is discussed in relation to the mechanisms incorporated in mathematical models as well as the role of stochasticity and determinism at cellular and tissue levels which resemble typical statistical and thermodynamic effects in physics.

Boundedness of the nominal coefficients in Gaussian RBF neural networks

The paper analyzes the boundedness of the coefficients involved in Gaussian expansion series. These series arise from the reconstruction of bandlimited functions, applying the sampling theorem with Gaussians as reconstruction filters. The boundedness of the ideal coefficients is a previous requirement that should be imposed to the approximation function. This is due to the fact that the coefficient sequence should be absolutely summable. With this sort of requirements, the targeted function is guaranteed to exhibit finite energy so that it will be manageable from the viewpoint of the approximation theory. On the other hand, the bounds of the coefficients affect considerably to the approximation errors and consequently to the accuracy of the estimation. The major result of this work is formalized in a series of propositions where it is stated how the coefficients are upper bounded by a signal “sinus cardinalis” (sinc). Finally, an energy measure of the approximation error is determined as a mean square error. In this line, a number of results are presented in both the univariate and the multivariate case showing how these errors strongly depend on the coefficients in the Gaussian expansion.