Stable and unbiased flow turbulence estimation from pulse echo ultrasound

A new method for stable and unbiased flow turbulence estimation has been developed for medical ultrasonic color flow imaging. Conventional turbulence estimates from a finite number of transmitted pulses could be biased, unreliable, and erroneous. We found that a conventional method cannot provide quantitative estimates of variance of flow velocity. We propose a new approach for flow turbulence estimation that is based on analysis of the flow velocity vectors. The new method estimates the variance of the flow velocity and provides reliable estimates for flow turbulence. Numerical examples, computer simulations, and experiments using a flow phantom demonstrate that the new method can estimate variance of flow velocity accurately and without bias. This work also reports a complete derivation in the time domain for both unbiased velocity and turbulence estimations. The results include two velocity estimation equations agreeing with the 1-D and 2-D autocorrelation methods derived from the frequency domain. The results indicate that the new method for flow turbulence is particularly useful when the 2-D autocorrelation method is used for color flow imaging. The new method also appears to be able to detect low turbulence; therefore, it may be useful for diagnosing abnormalities such as minor stenoses and valvular jets.     

Characterization of Frequency Stability: Uncertainty Due to the Autocorrelation of the Frequency Fluctuations

We considered the general sampling form for the estimate of the Allan variance which is the proposed measure of frequency stability in the time domain, and we defined a variable proportional to the difference between the average fractional frequency fluctuations over the time interval ? to derive the autocorrelation coefficient of the process to which the variable belongs. Calculations of the variance of the estimated Allan variance proved that it may be convergent to its true value with infinite sample number for considered spectral densities of frequency noise. We also applied the results to estimations of frequency measurements to know the influence of the autocorrelation of the process considered. In order to obtain some direct estimates of the confidence of the estimate, distributions of the estimate were plotted by means of computer simulations, and were compared with the chi-square distribution. Those results suggested that for white-and flicker-phase noises (and white-frequency noise) we have to take into account the autocorrelation of the process, while for flicker-or random-walk-frequency noise we may regard the process as a nearly independent (and Gaussian) one.     

An exact closed-form solution for linear multi-degree-of-freedom systems under Gaussian white noise via the Wiener path integral technique

The exact joint response transition probability density function (PDF) of linear multi-degree-of-freedom oscillators under Gaussian white noise is derived in closed-form based on the Wiener path integral (WPI) technique. Specifically, in the majority of practical implementations of the WPI technique, only the first couple of terms are retained in the functional expansion of the path integral related stochastic action. The remaining terms are typically omitted since their evaluation exhibits considerable analytical and computational challenges. Obviously, this approximation affects, unavoidably, the accuracy degree of the technique. However, it is shown herein that, for the special case of linear systems, higher than second order variations in the path integral functional expansion vanish, and thus, retaining only the first term (most probable path approximation) yields the exact joint response transition Gaussian PDF. Both single- and multi-degree-of-freedom linear systems are considered as illustrative examples for demonstrating the exact nature of the derived solutions. In this regard, the herein derived analytical results are also compared with readily available in the literature closed-form exact solutions obtained by alternative stochastic dynamics techniques. In addition to the mathematical merit of the derived exact solution, the closed-form joint response transition PDF can also serve as a benchmark for assessing the performance of alternative numerical solution methodologies.     

Joint statistics of combined first- and second-order random processes

Two-term Volterra series are often used to describe non-linear random processes in subject areas as diverse as communication theory and the dynamics of offshore structures. It is well known that the characteristic function, the moments and the cumulants of such processes can be calculated analytically and that the probability density function can be calculated accurately and efficiently. This paper considers the joint statistics between two such random processes by deriving: (i) an exact expression for the joint characteristic function; (ii) an efficient means for calculating the joint moments; (iii) an ‘exact’ numerical means for calculating the joint probability density function (jpdf). For the special case of a combined first- and second-order process and a pure first-order process it is shown that it is possible to derive analytical expressions for the characteristic function and to calculate the jpdf accurately and efficiently using saddle-point integration. In addition to the above, the maximum entropy principle is used to calculate the jpdf.     

Estimation of stochastic environment force for master–slave robotic system

The aim of this work is to obtain the maximum likelihood estimate (MLE) of controller-gain parameters \(\widehat{K} \) of the slave robot to determine the stochastic environment force. This is accomplished by measuring the joint positions of master and slave for a known master torque using stochastic difference equation. Here, the environmental force is modelled as a zero-mean white Gaussian random process. Therefore, the joint probability distribution function (pdf) of the slave angle over a given time duration can be computed as a function of the parameters ‘K’. This pdf is maximized with respect to ‘K’ to obtain the MLE of controller-gain parameters. Subsequently, convergence analysis of error in the estimates is performed. Also, an expression of the Cramer–Rao lower bound (CRLB) is derived to measure accuracy of the estimation. Comparison of CRLB with variance of MLE supports that our estimates are asymptotically efficient. The estimation performance is validated analytically and through simulations carried out on a two-link master–slave robotic system.     

A copula-based Gaussian mixture closure method for stochastic response of nonlinear dynamic systems

Gaussian closure method is commonly used in the analysis of nonlinear stochastic systems. However, Gaussian closure may lead to unacceptable errors when system response is very much different from being Gaussian, and accuracy of the method decreases as the nonlinearity of the system increases. The need for better accuracy in strongly non-linear problems has caused the development of non-Gaussian closure schemes. In this paper, we develop a new copula-based Gaussian mixture closure method for randomly excited nonlinear systems. Our method relies on the assumption of marginal PDF of response in terms of finite Gaussian mixture model, and the derivation of joint PDF with aid of dependence modeling of Gaussian copula. By substituting the non-Gaussian PDF representation into moment equations of nonlinear system, we further develop an optimization-based closure scheme for the solution of the unknown parameters in joint PDF. In this way, PDF and thus, moments of response of highly nonlinear system can be described in a more flexible and robust way. Effectiveness of the new closure method is demonstrated by a nonlinear and a Duffing oscillator that are subjected to Gaussian white noise. The results are compared with the Gaussian closure and exact solution. It has been shown that Gaussian closure is a special case of the new closure method, and accuracy of Gaussian closure is the lower bound of that of the new closure method.     

Non-linear stochastic control via stationary response design

This paper presents a strategy for controlling externally excited stochastic systems with uncertain parameters. The control objective is to drive the system response from an arbitrary initial distribution to a prescribed stationary probability density function (PDF). This problem can be interpreted as the stochastic stabilization about a PDF. The control consists of a non-linear feedback part and a switching term inspired by robust sliding control concepts. The control of a one-dimensional stochastic process with a Gaussian target-PDF is used to illustrate the approach. The control performance is evaluated by studying the time evolution of the first and second order moments of the response. The dependence of the response on the number of feedback terms and the rate of convergence to the stationary PDF are studied numerically. Motivated by the fast convergence observed, a feedback controller with time-varying gains is applied to the problem of tracking a moving PDF. Monte Carlo simulations validate very well the control for both stabilization and tracking problems.     

Bias of mean value and mean square value measurements based onquantized data

This paper investigates the imperfect fulfilment of the validity conditions of the noise model quantization. The general expressions of the deviations of the moments from Sheppard's corrections are derived. Approximate upper and lower bounds of the bias are given for the measurement of first- and second-order moments of sinusoidal, uniformly distributed, and Gaussian signals. It is shown that because of the uncontrollable mean value at the input of the ADC (offset, drift), the worst-case values have to be investigated; it is illustrated how a simple-form envelope function of the errors can be used as an upper bound. Since the worst-case relative positions of the signal and the quantization characteristics are taken into account, the results are valid for both midtread and midrise quantizers, while in the literature results are given for a selected quantizer type only     

Evaluating systematic differences between laboratories in interlaboratory comparisons

A Bayesian analysis of the data from interlaboratory comparisons involving a single stable traveling standard is presented. The approach is based on the assumption that each participating laboratory provides an estimate of the value of the measurand with zero estimated bias. In addition, it is assumed that each of the reported uncertainties is given in the form of two separate components, one associated with random effects and the other associated with systematic effects. It is finally assumed that all information is consistent. Using Gaussian probability density functions, simple formulas for the joint estimate of the value of the measurand and for the a posteriori estimates of the biases and of their differences are derived. Formulas for the uncertainties of all these estimates are also given.     

基于高斯混合模型的非高斯随机振动幅值概率密度函数

针对非高斯振动信号的幅值概率密度函数难以用数学模型表述的问题,提出了基于高斯混合模型的非高斯概率密度函数表示方法。首先,基于时域样本信号得到非高斯振动信号的高阶矩估计值。其次,基于高斯随机过程偶次高阶矩之间的定量关系,结合二阶高斯混合模型建立方程组,求解得到混合模型中每个高斯分量的方差和权值。然后,将各高斯分量的权值和方差代入高斯混合模型,得到适用于对称非高斯振动信号的幅值概率密度函数。最后,通过仿真信号和实测振动信号,验证了该方法的有效性和适用性。     

Pseudophase information from the complex analytic signal of speckle fields and its applications. Part II: statistical properties of the analytic signal of a white-light speckle pattern applied to the microdisplacement measurement

To provide a theoretical background for the superiority of the signal-domain phase-only correlation (SDPOC) technique proposed here for microdisplacement measurement, we study the first- and the second-order statistical properties of the complex amplitude of an analytic signal of a white-light speckle pattern, under the assumption of a Gaussian random process, and give a formula for the autocorrelation function of the pseudophase associated with the complex analytic signal. Based on these results, we show mathematically that SDPOC has a performance advantage over conventional intensity-based correlation techniques.     

Bayesian time–domain approach for modal updating using ambient data

The problem of identification of the modal parameters of a structural model using measured ambient response time histories is addressed. A Bayesian time–domain approach for modal updating is presented which is based on an approximation of a conditional probability expansion of the response. It allows one to obtain not only the optimal values of the updated modal parameters but also their associated uncertainties, calculated from their joint probability distribution. Calculation of the uncertainties of the identified modal parameters is very important if one plans to proceed in a subsequent step with the updating of a theoretical finite-element model based on modal estimates. The proposed approach requires only one set of response data. It is found that the updated PDF can be well approximated by a Gaussian distribution centered at the optimal parameters at which the updated PDF is maximized. Examples using simulated data are presented to illustrate the proposed method.     

A univariate dimension-reduction method for multi-dimensional integration in stochastic mechanics

This paper presents a new, univariate dimension-reduction method for calculating statistical moments of response of mechanical systems subject to uncertainties in loads, material properties, and geometry. The method involves an additive decomposition of a multi-dimensional response function into multiple one-dimensional functions, an approximation of response moments by moments of single random variables, and a moment-based quadrature rule for numerical integration. The resultant moment equations entail evaluating N number of one-dimensional integrals, which is substantially simpler and more efficient than performing one N-dimensional integration. The proposed method neither requires the calculation of partial derivatives of response, nor the inversion of random matrices, as compared with commonly used Taylor expansion/perturbation methods and Neumann expansion methods, respectively. Nine numerical examples involving elementary mathematical functions and solid-mechanics problems illustrate the proposed method. Results indicate that the univariate dimension-reduction method provides more accurate estimates of statistical moments or multidimensional integration than first- and second-order Taylor expansion methods, the second-order polynomial chaos expansion method, the second-order Neumann expansion method, statistically equivalent solutions, the quasi-Monte Carlo simulation, and the point estimate method. While the accuracy of the univariate dimension-reduction method is comparable to that of the fourth-order Neumann expansion, a comparison of CPU time suggests that the former is computationally far more efficient than the latter.     

Numerical solutions of Fokker-Planck equation of nonlinear systems subjected to random and harmonic excitations

A new approach for an efficient numerical implementation of the path integral (PI) method based on non-Gaussian transition probability density function (PDF) and the Gauss-Legendre integration scheme is developed. This modified PI method is used to solve the Fokker-Planck (FP) equation and to study the nature of the stochastic and chaotic response of the nonlinear systems. The steady state PDF, periodicity, jump phenomenon, noise induced changes in joint PDF of the states are studied by the modified PI method. A computationally efficient higher order, finite difference (FD) technique is derived for the solution of higher-dimensional FP equation. A two degree of freedom nonlinear system having Coulomb damping with a variable friction coefficient subjected to Gaussian white noise excitation is considered as an example which can represent a bladed disk assembly of turbo-machinery blades. Effects of normal force and viscous damping on the mean square response are investigated.     

Dependence of the dose estimate on the time pattern of intake by the example of tritiated water intakes

The uncertainties related to activity measurement and time pattern of intake in routine monitoring of internal exposure are considered through the example of tritiated water intakes. For this purpose, a combination of intake-to-bioassay and bioassay-to-intake calculations with Monte Carlo integration technique is introduced as a method of investigation. The time pattern of intake and the measured activity are defined as random input quantities. The probability density functions (PDFs) of the input quantities are defined and a Monte Carlo integration is performed to obtain the PDF of the output quantity which is either the value of intake estimated from a measured value of activity or the estimated activity from a given value of intake. Different possible estimates of the intake are considered: some represent the parameters of the PDF of the output quantity, others are derived from the commonly used constant chronic, I(CC), and mid-point, I(1/2), methods. The combinations of activity and intake estimates that would provide a stable estimate of the initial intake in intake-to-bioassay and bioassay-to-intake calculations were studied. Several intake estimates satisfying this requirement can be chosen depending on the task to be solved by adjusting the proper activity estimate.     

Probabilistic bounded non-linear control algorithm for linear structures

Non-linear control algorithms with limited control force has been widely explored and has shown promising results, but the probabilistic analysis of such control algorithms has been made restrictively due to their non-linear nature and corresponding difficulty in obtaining analytical solution of the joint probability density function (PDF). In this paper, the method for the probabilistic analysis on the bounded non-linear control algorithm is proposed based on the equivalent non-linear system method and additional regression analysis. Numerical examples show that the proposed approximate joint PDF of the closed-loop system subjected to a Gaussian white noise and a Kanai–Tajimi filtered Gaussian white noise matches closely with the joint PDF obtained statistically. The effectiveness of the bounded non-linear control is also investigated utilizing the calculated approximate joint PDF. Time history analysis results indicate that the same control performance level as the linear controller is achieved when 50% of the maximum control force of the linear controller is used for the non-linear controller.     

Radial basis function neural networks solution for stationary probability density function of nonlinear stochastic systems

A radial basis function neural networks (RBF-NN) solution of the reduced Fokker–Planck-Kolmogorov (FPK) equation is proposed in this paper. The activation functions consist of normalized Gaussian probability density functions (PDFs). The use of normalized Gaussian PDFs leads to a simple constraint on the coefficients for normalization of the RBF-NN solution, which as a constraint is imposed with the help of the method of Lagrange multiplier. The relationship between the proposed RBF-NN PDF solution and the generalized cell mapping with short-time Gaussian approximation is discussed, which provides a justification for Gaussian PDFs with varying means and variances in the state space. The optimal number of neurons or activation functions, which leads to the smallest error, is investigated. Four examples are presented to show the effectiveness of the proposed solution method. The results indicate that the proposed solution method is a very efficient and accurate way to compute the stationary PDF of nonlinear stochastic systems. It is also found that the distribution of the optimal coefficients as a function of the mean of Gaussian activation functions is similar to the steady-state PDF solution. Finally, we should point out that an important advantage of the RBF-NN method over methods such as finite element and finite difference is its ability to obtain solutions of the FPK equation for multi-degree-of-freedom stochastic systems.     

典型非高斯分布假设下累积和检测方法性能研究

累积和检测方法是根据声呐目标信号出现与消失时概率密度函数（Probability Distribution Function,PDF）的变化进行有效的瞬态信号检测。以非高斯模型t分布假设替代传统的高斯分布方差变化假设作为描述瞬态信号的PDF形式,推导了累积和检验统计量的表达、更新量PDF求取的数值方法,利用快速傅里叶变换法计算了门限和自由度等检测参数。利用仿真的落水信号、船体加速信号和消声水池实验数据进行检验。结果表明,基于t分布假设的累积和方法对瞬态脉冲信号的检测效果优于常规累积和方法,能更快地响应信号变化,更好地抑制背景干扰。     

一种基于累积量的ESPRIT型盲波束形成算法

讨论一种用于非高斯信号源方位估计的盲波束形成算法。假设背景噪声是二阶统计未知的有色(空间相关)高斯噪声,基于“阵元对”模型的ESPRIT方位估计算法可以通过用累积量矩阵取代自相关矩阵完成信号空间的重建。经过这种处理,加性有色高斯噪声可被滤除,因此算法不需要相关噪声矩阵的知识。当加性噪声源是空间相关矩阵未知的有色高斯噪声时,计算机仿真比较了基于累积量的ESPRIT算法与基于二阶统计的ESPRIT算法的性能。