Observations on the McKay I Bessel distribution II

Performance of time-frequency representation techniques to measure blood flow turbulence with pulsed-wave Doppler ultrasound

This article investigates basic properties of the Bessel distribution, a power series distribution which has not been fully explored before. Links with some well-known distributions such as the von Mises-Fisher distribution are described. A simulation scheme is also proposed to generate random samples from the Bessel distribution. This scheme is useful in Bayesian inferences and Monte Carlo computation.

A kernel based on the first kind Bessel function of order one is proposed to compute the time-frequency distributions of nonstationary signals. This kernel can suppress the cross terms of the distribution effectively. It is shown that the Bessel distribution (the time-frequency distribution using Bessel kernel) meets most of the desirable properties with high time-frequency resolution. A numerical alias-free implementation of the distribution is presented. Examples of applications in time-frequency analysis of the heart's sound and Doppler blood flow signals are given to show that the Bessel distribution can be easily adapted to two very different signals for cardiovascular signal processing. By controlling a kernel parameter, this distribution can be used to compute the time-frequency representations of transient deterministic and random signals. The study confirms the potentials of the proposed distribution in nonstationary signal analysis.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Classification of lower limb arterial stenoses from doppler blood flow signal analysis with time-frequency representation and pattern recognition techniques

Using Reduced Interference Distribution to Analyze Abnormal Cardiac SignalDue to the non-stationary, multicomponent nature of biomedical signals, the use of time-frequency analysis can be inevitable for these signals. The choice and selection of the proper Time-Frequency Distribution (TFD) that can reveal the exact multicomponent structure of biological signals is vital in many applications, including the diagnosis of medical abnormalities. In this paper, the instantaneous frequency techniques using two distribution functions are applied for analysis of biological signals. These distributions are the Wigner-Ville Distribution and the Bessel Distribution. The simulation performed on normaland abnormal cardiac signals show that the Bessel Distribution can clearly detect the QRS complexes. However, Wigner-Ville Distribution was able to detect the QRS complexes in the normal signa, but fails to detect these complexes in the abnormal cardiac signal.

The Bessel distribution, introduced recently by Yuan and Kalbfleisch (Ann. Inst. Math. Statist., 2000), can be useful in many applications. In particular, this distribution appears in two Bayesian estimation problems, namely, estimation of the noncentrality parameter of a noncentral chi-square distribution and of the parameters of Downton's bivariate exponential distribution. Implementation of Markov chain Monte Carlo algorithms requires generation of observations from the Bessel distribution. In this paper we propose and compare exact simulation schemes generating Bessel variates based on certain properties of the distribution as well as the rejection method.

FAST CALCULATION OF THE BESSEL TIME-FREQUENCY DISTRIBUTION

The time-frequency distribution (TFD) of Doppler blood flow signals is usually obtained using the spectrogram, which requires signal stationarity and is known to produce large estimation variance. This paper examines four alternative, nonstationary spectral estimators: a smoothed pseudo-Wigner distribution (SPWD), the Choi-Williams distribution (CWD), the Bessel distribution (BD), and the novel, adaptive constant-Q distribution (AQD) for their applicability to Doppler ultrasound. A synthetic Doppler signal, simulating the nonaxial and pulsatile flow of the common carotid artery, was used for quantitative comparisons at different signal-to-noise-ratios (SNR) of 0, 10, 20, and 30 dB as well as noise free. The cross-correlation (rho) and the root-mean-square-error (RMSE) were calculated after log-compression for each technique and SNR relative to the theoretical distribution. The AQD consistently had the lowest RMSE (< or =53.7%) and the highest rho (> or =0.889) of all the TFDs, irrespective of the SNR. The SPWD performed better than the spectrogram, which performed better than the BD and the CWD. Qualitative comparisons were carried out using in vivo data acquired with a 10 MHz ultrasound cuff transducer positioned around the distal aorta of a rabbit. In vivo, the AQD was considered best with respect to background noise and internal gray scale features; it was rated second (after the spectrogram) in depicting the spectral envelope. The AQD performed better as a Doppler spectral estimator than the traditional spectrogram and the other TFDs under the conditions studied here.

We show that the Bessel distribution attached to a generic representation of GL(3, F), where F is a p-adic field, is given by a Bessel function.

We propose a novel algorithm for extracting atrial activity from single lead electrocardiogram (ECG) signal sustained with atrial fibrillation (AF), based on a short-time expansion of an orthogonal basis function set. The method preserves the time variation of spectral content of the underlying AF signal, thus time-frequency analysis of the AF signal can be successfully performed. The new method is compared to the standard average beat subtraction (ABS) method using synthetic AF sustained ECG data. The orthogonal basis expansion method has a higher correlation with the original AF signal compared to the ABS method for a range of signal to noise ratio (SNR) levels, and correlation is improved by 16% at an SNR of 0dB. Time-frequency analysis of the reconstructed AF signal based on Bessel distribution also shows the superiority of the orthogonal basis expansion method over ABS.

The generalized Kaiser–Bessel window function is defined via the modified Bessel function of the first kind and arises frequently in tomographic image reconstruction. In this paper, we study in details the properties of the Kaiser–Bessel distribution, which we define via the symmetric form of the generalized Kaiser–Bessel window function. The Kaiser–Bessel distribution resembles to the Bessel distribution of McKay of the first type, it is a platykurtic or sub-Gaussian distribution, it is not infinitely divisible in the classical sense and it is an extension of the Wigner’s semicircle, parabolic and n-sphere distributions, as well as of the ultra-spherical (or hyper-spherical) and power semicircle distributions. We deduce the moments and absolute moments of this distribution and we find its characteristic and moment generating function in two different ways. In addition, we find its cumulative distribution function in three different ways and we deduce a recurrence relation for the moments and absolute moments. Moreover, by using a formula of Ismail and May on quotient of modified Bessel functions of the first kind, we deduce a closed-form expression for the differential entropy. We also prove that the Kaiser–Bessel distribution belongs to the family of log-concave and geometrically concave distributions, and we study in details the monotonicity and convexity properties of the probability density function with respect to the argument and each of the parameters. In the study of the monotonicity with respect to one of the parameters we complement a known result of Gronwall concerning the logarithmic derivative of modified Bessel functions of the first kind. Finally, we also present a modified method of moments to estimate the parameters of the Kaiser–Bessel distribution, and by using the classical rejection method we present two algorithms for sampling independent continuous random variables of Kaiser–Bessel distribution. The paper is closed with conclusions and proposals for future works.

Modeling the temperature profile of an extrudate in material extrusion additive manufacturing

The time-frequency distribution of the Doppler ultrasound blood flow signal is normally computed by using the short-time Fourier transform or autoregressive modeling. These two techniques require stationarity of the signal during a finite interval. This requirement imposes some limitations on the distribution estimate. In the present study, three new techniques for nonstationary signal analysis (the Choi-Williams distribution, a reduced interference distribution, and the Bessel distribution) were tested to determine their advantages and limitations for analysis of the Doppler blood flow signal of the femoral artery. For the purpose of comparison, a model stimulating the quadrature Doppler signal was developed, and the parameters of each technique were optimized based on the theoretical distribution. Distributions computed using these new techniques were assessed and compared with those computed using the short-time Fourier transform and autoregressive modeling. Three indexes, the correlation coefficient, the integrated squared error, and the normalized root-mean-squared error of the mean frequency waveform, were used to evaluate the performance of each technique. The results showed that the Bessel distribution performed the best, but the Choi-Williams distribution and autoregressive modeling are also techniques which can generate good time-frequency distributions of Doppler signals.

Positive characteristics have been found for diffuse, striationfree, 1-20 ma dc, positive columns in 150-mm Xe, Kr, or Ar, with 0.1% ${\mathrm{N}}_{2}$ in 10-cm tubes. With Xe+${\mathrm{N}}_{2}$, a continuous spectrum is emitted; with other rare gases selections of ${\mathrm{N}}_{2}$ bands. The positiveness of the characteristic results from (1) the disappearance of ions principally by dissociative recombination and (2) ionization which is in effect single stage. For similar current densities the gradient varies little with tube diameter. Probe measurements indicate that the electron and ion density has a relatively flat distribution over a large central part of the tube as compared with the Bessel distribution; the latter resulting from ion loss to the walls by ambipolar diffusion. This broadening out of the discharge increases with the current. For steady convection-free discharges where volume recombination and diffusion are both contributing to ion loss, the degree of should be governed by a constriction number $C$ given by $C=\frac{{D}_{a}}{\ensuremath{\alpha}{R}^{2}{n}_{e}}$, where ${D}_{a}$ is the ambipolar diffusion constant, $\ensuremath{\alpha}$ is the recombination coefficient (supposed constant everywhere), $R$ is the tube radius, and ${n}_{e}$ the ion concentration.When the current is increased beyond a critical value, the discharge changes abruptly to a filamentary form, the same whether ${N}_{2}$ is present or not, having a several-fold lower gradient, a negative characteristic, and emitting mainly the line spectrum of the rare gas. This discharge is believed to be diffusion and convection controlled, the (atomic) ions diffusing from the hot core to the cooler periphery where they form molecular ions and recombine dissociatively.Such a mechanism as the above, involving a temperature gradient favoring the existence of molecular ions and dissociative recombination in the outer regions is believed to account quite generally for greater than that corresponding to the Fabrikant-Spenke curve.

ABSTRACTIn survival analysis and reliability studies, problems with random sample size arise quite frequently. More specifically, in cancer studies, the number of clonogens is unknown and the time to relapse of the cancer is defined by the minimum of the incubation times of the various clonogenic cells. In this article, we have proposed a new model where the distribution of the incubation time is taken as Weibull and the distribution of the random sample size as Bessel, giving rise to a Weibull–Bessel distribution. The maximum likelihood estimation of the model parameters is studied and a score test is developed to compare it with its special submodel, namely, exponential–Bessel distribution. To illustrate the model, two real datasets are examined, and it is shown that the proposed model, presented here, fits better than several other existing models in the literature. Extensive simulation studies are also carried out to examine the performance of the estimates.