Efficient data assimilation for spatiotemporal chaos: A local ensemble transform Kalman filter

Data assimilation is an iterative approach to the problem of estimating the state of a dynamical system using both current and past observations of the system together with a model for the system’s time evolution. Rather than solving the problem from scratch each time new observations become available, one uses the model to “forecast” the current state, using a prior state estimate (which incorporates information from past data) as the initial condition, then uses current data to correct the prior forecast to a current state estimate. This Bayesian approach is most effective when the uncertainty in both the observations and in the state estimate, as it evolves over time, are accurately quantified. In this article, we describe a practical method for data assimilation in large, spatiotemporally chaotic systems. The method is a type of “ensemble Kalman filter”, in which the state estimate and its approximate uncertainty are represented at any given time by an ensemble of system states. We discuss both the mathematical basis of this approach and its implementation; our primary emphasis is on ease of use and computational speed rather than improving accuracy over previously published approaches to ensemble Kalman filtering. We include some numerical results demonstrating the efficiency and accuracy of our implementation for assimilating real atmospheric data with the global forecast model used by the US National Weather Service.     

基于Lorenz模型的集合预报与单一预报的比较研究

根据非线性局部Lyapunov向量方法和增长模繁殖方法,选取Lorenz63模型和Lorenz96模型的不同状态为例,对集合预报与单一预报的预报技巧开展了对比研究.结果表明:与单一预报比较,集合预报的均方根误差和型异常相关有明显改善,随预报时间推移,改善效果越显著,且集合平均优于单一预报的实验个例数逐渐增多.就概率分布(f)而言,单一预报状态的f与真实状态基本一致,不随时间变化;而集合平均预报状态的f则随时间呈现出值域变窄、峰值变大的特点.表明随预报时间的延长,单一预报状态为混沌吸引子上的随机状态,而集合平均预报状态为吸引子子集上的随机状态,这可能是集合平均误差小于单一预报的原因.     

Exploring the need for localization in ensemble data assimilation using a hierarchical ensemble filter

Good performance with small ensemble filters applied to models with many state variables may require ‘localizing’ the impact of an observation to state variables that are ‘close’ to the observation. As a step in developing nearly generic ensemble filter assimilation systems, a method to estimate ‘localization’ functions is presented. Localization is viewed as a means to ameliorate sampling error when small ensembles are used to sample the statistical relation between an observation and a state variable. The impact of spurious sample correlations between an observation and model state variables is estimated using a ‘hierarchical ensemble filter’, where an ensemble of ensemble filters is used to detect sampling error. Hierarchical filters can adapt to a wide array of ensemble sizes and observational error characteristics with only limited heuristic tuning. Hierarchical filters can allow observations to efficiently impact state variables, even when the notion of ‘distance’ between the observation and the state variables cannot be easily defined. For instance, defining the distance between an observation of radar reflectivity from a particular radar and beam angle taken at 1133 GMT and a model temperature variable at 700 hPa 60 km north of the radar beam at 1200 GMT is challenging. The hierarchical filter estimates sampling error from a ‘group’ of ensembles and computes a factor between 0 and 1 to minimize sampling error. An a priori notion of distance is not required. Results are shown in both a low-order model and a simple atmospheric GCM. For low-order models, the hierarchical filter produces ‘localization’ functions that are very similar to those already described in the literature. When observations are more complex or taken at different times from the state specification (in ensemble smoothers for instance), the localization functions become increasingly distinct from those used previously. In the GCM, this complexity reaches a level that suggests that it would be difficult to define efficient localization functions a priori. There is a cost trade-off between running hierarchical filters or running a traditional filter with larger ensemble size. Hierarchical filters can be run for short training periods to develop localization statistics that can be used in a traditional ensemble filter to produce high quality assimilations at reasonable cost, even when the relation between observations and state variables is not well-known a priori. Additional research is needed to determine if it is ever cost-efficient to run hierarchical filters for large data assimilation problems instead of traditional filters with the corresponding total number of ensemble members.     

Correlations without joint distributions in quantum mechanics

The use of joint distribution functions for noncommuting observables in quantum thermodynamics is investigated in the light of L. Cohen's proof that such distributions are not determined by the quantum state. Cohen's proof is irrelevant to uses of the functions that do not depend on interpreting them as distributions. An example of this, from quantum Onsager theory, is discussed. Other uses presuppose that correlations betweenp andq values depend at least on the state. But correlations may be fixed by the state even though the distribution varies from one ensemble to another represented by that state. Taking covariance as a measure of correlation, it is shown that the different commonly used joint distributions yield the same correlations for a given state. A general characterization is given for a family of distributions with this same covariance.     

Design and analysis of a BLPC vocoder-based adaptive feedback cancellation with probe noise

The band-limited linear predictive coding (BLPC) vocoder-based adaptive feedback cancellation (AFC) removes the high-frequency bias, while the low frequency bias persists between the desired input signal and the loudspeaker signal in the estimate of the feedback path. In this paper, we present a BLPC vocoder-based adaptive feedback canceller with probe noise with an objective of reducing the low-frequency bias in digital hearing-aids. A step-wise mathematical analysis of the proposed feedback canceller is presented employing the recursive least square and normalized least mean square adaptive algorithms. It is observed that the optimal solution of the feedback path is unbiased for an unshaped probe noise, but is biased for a shaped probe signal; the bias term does not consist of correlation between the desired input and the loudspeaker output. The identifiability conditions are analysed and it is shown that a delay, greater than or equal to the length of the adaptive filter, must be introduced in the forward path to achieve an unbiased feedback path estimate. Algorithm analysis and computer simulations presented in this paper justify the reason for selecting the proposed design over the existing BLPC vocoder-based feedback cancellation algorithm.     

Optimization of the algorithms for estimating the ultrasonic attenuation along the propagation path

In this study, we perform statistical analysis on two methods used to estimate the total ultrasound attenuation along the propagation path from the surface of the transducer to a region of interest at a particular depth; namely, the spectral-fit method and the multiple-filter method. We derive mathematical equations for the bias and variance in the attenuation estimates as a function of region of interest (ROI) size, imaging system bandwidth, and number of independent Gaussian filters (for the multiple filter method). We use numerical simulations to validate the mathematical equations and compare the two algorithms. The results show that the variance in the total attenuation coefficient estimates obtained with the two methods are comparable, and that the estimates are unbiased. For the multiple filter method, the optimal number of Gaussian filters is two.     

基于Khatri-Rao子空间和传播算子的宽带声源波达方向估计算法

本文提出基于Khatri-Rao子空间和传播算子的宽带声源波达方向估计算法。该算法将声源不同频率处的协方差矩阵变换重排为一个高维矩阵,然后利用传播算子方法估计宽带声源波达方向。该算法计算复杂度介于聚焦Khatri-Rao子空间和相干子空间算法之间。仿真和实验结果表明,该算法在降低计算量的同时,估计误差与聚焦Khatri-Rao子空间算法相近,远小于相干子空间算法。     

Ensemble Kalman filter with the unscented transform

A modification scheme to the ensemble Kalman filter (EnKF) is introduced based on the concept of the unscented transform [S. Julier, J. Uhlmann, H. Durrant-Whyte, A new method for the nonlinear transformation of means and covariances in filters and estimators, IEEE Trans. Automat. Control. 45 (2000) 477-482; S.J. Julier, J.K. Uhlmann, Unscented filtering and nonlinear estimation, Proc. IEEE 92 (2004) 401-422], which therefore will be called the ensemble unscented Kalman filter (EnUKF) in this work. When the error distribution of the analysis is symmetric (not necessarily Gaussian), it can be shown that, compared with the ordinary EnKF, the EnUKF has more accurate estimations of the ensemble mean and covariance of the background by examining the multidimensional Taylor series expansion term by term. This implies that, the EnUKF may have better performance in state estimation than the ordinary EnKF in the sense that the deviations from the true states are smaller. For verification, some numerical experiments are conducted on a 40-dimensional system due to Lorenz and Emanuel [E.N. Lorenz, K.A. Emanuel, Optimal sites for supplementary weather observations: Simulation with a small model, J. Atmos. Sci. 55 (1998) 399-414]. Simulation results support our argument.     

Quantum Mechanical Nature in Liquid NMR Quantum Computing

The quantum nature of bulk ensemble NMR quantum computing the center of recent heated debate,is addressed. Concepts of the mixed state and entanglement are examined, and the data in a two-qubit liquid NMRquantum computation are analyzed. The main points in this paper are: i) Density matrix describes the “state“ of anaverage particle in an ensemble. It does not describe the state of an individual particle in an ensemble; ii) Entanglementis a property of the wave function of a microscopic particle (such as a molecule in a liquid NMR sample), and separabilityof the density matrix cannot be used to measure the entanglement of mixed ensemble; iii) The state evolution in bulk-ensemble NMRquantum computation is quantum-mechanical; iv) The coefficient before the effective pure state densitymatrix, e, is a measure of the simultaneity of the molecules in an ensemble. It reflects the intensity of the NMR signaland has no significance in quantifying the entanglement in the bulk ensemble NMR system. The decomposition of thedensity matrix into product states is only an indication that the ensemble can be prepared by an ensemble with theparticles unentangled. We conclude that effective-pure-state NMR quantum computation is genuine, not just classicalsimulations.     

观测站非机动情况下的渐进无偏目标跟踪算法

针对无源目标跟踪中将非线性测量方程伪线性化带来有偏估计的缺点,提出了一种基于方位和径向速度联合估计的渐进无偏目标跟踪算法。该算法通过对方位-径向速度组成的伪线性测量方程的噪声协方差矩阵进行约束,使其均方误差达到最小,得到目标状态向量的最小二乘无偏解。仿真和实验结果表明,在观测站非机动情况下,算法能够渐进无偏地收敛到真值,收敛后的距离跟踪误差在5%以内。     

Random matrix theory filters in portfolio optimisation: A stability and risk assessment

Random matrix theory (RMT) filters, applied to covariance matrices of financial returns, have recently been shown to offer improvements to the optimisation of stock portfolios. This paper studies the effect of three RMT filters on the realised portfolio risk, and on the stability of the filtered covariance matrix, using bootstrap analysis and out-of-sample testing.We propose an extension to an existing RMT filter, (based on Krzanowski stability), which is observed to reduce risk and increase stability, when compared to other RMT filters tested. We also study a scheme for filtering the covariance matrix directly, as opposed to the standard method of filtering correlation, where the latter is found to lower the realised risk, on average, by up to 6.7%.We consider both equally and exponentially weighted covariance matrices in our analysis, and observe that the overall best method out-of-sample was that of the exponentially weighted covariance, with our Krzanowski stability-based filter applied to the correlation matrix. We also find that the optimal out-of-sample decay factors, for both filtered and unfiltered forecasts, were higher than those suggested by Riskmetrics [J.P. Morgan, Reuters, Riskmetrics technical document, Technical Report, 1996. http://www.riskmetrics.com/techdoc.html], with those for the latter approaching a value of α=1.In conclusion, RMT filtering reduced the realised risk, on average, and in the majority of cases when tested out-of-sample, but increased the realised risk on a marked number of individual days–in some cases more than doubling it.     

Universal adaptive lattice filters. Adaptation for a given root of the estimating correlation matrix

Algorithms for the adjustment of adaptive lattice filters according to a given root of the estimating noise-correlation matrix (CM) are considered. A basic algorithm is synthesized from which can be derived adjustment algorithms that take into account a priori information on the CM structure. Methods for simplification of the algorithm and increasing its efficiency are examined.     

基于高斯过程的混沌时间序列单步与多步预测

针对混沌时间序列单步和多步预测,提出基于复合协方差函数的高斯过程 (GP)模型方法.GP模型的确立由协方差函数决定,通过对训练数据集的学习,在证据最大化框架内,利用矩阵运算和优化算法自适应地确定协方差函数和均值函数中的超参数.GP模型与神经网络、模糊模型相比,其可调整参数很少.将不同复合协方差函数的GP模型应用在混沌时间序列单步及多步提前预测中,并与单一协方差函数的GP、支持向量机、最小二乘支持向量机、径向基函数神经网络等方法进行了比较.仿真结果表明,基于不同复合协方差函数的GP方法能精确地预测混沌时间序 关键词： 高斯过程 混沌时间序列 预测 模型比较     

Covariance NMR spectroscopy by singular value decomposition

Covariance NMR is demonstrated for homonuclear 2D NMR data collected using the hypercomplex and TPPI methods. Absorption mode 2D spectra are obtained by application of the square-root operation to the covariance matrices. The resulting spectra closely resemble the 2D Fourier transformation spectra, except that they are fully symmetric with the spectral resolution along both dimensions determined by the favorable resolution achievable along omega2. An efficient method is introduced for the calculation of the square root of the covariance spectrum by applying a singular value decomposition (SVD) directly to the mixed time-frequency domain data matrix. Applications are shown for 2D NOESY and 2QF-COSY data sets and computational benchmarks are given for data matrix dimensions typically encountered in practice. The SVD implementation makes covariance NMR amenable to routine applications.     

多通道光谱采集系统滤光片设计方法研究

普通彩色相机加载宽带滤光片构造的多通道光谱采集系统的光谱重建性能与滤光片的选用密切相关。针对上述问题,提出基于主成分分析法的合成滤光片设计方法,旨在获得具有较高光谱重建性能并对所有图像场景均适用的最优滤光片组合。在收集多个宽带滤光片的基础上,测得其透射率并转换成矩阵形式;采用主成分分析方法提取该矩阵的前2个主成分,标准化处理后的主成分即为所求合成滤光片的透射率。为了验证上述方法获得的滤光片的性能,利用色差和光谱均方根误差2个指标对加载了合成滤光片的仿真采集系统的光谱重建精度进行了评价。实验结果表明:采用该方法得到的滤光片组合优于常用方法得到的滤光片组合, 用其构造的成像系统具有较高的色度重建精度和光谱重建精度,此外,改变目标场景颜色特性,其重建性能保持稳定。     

Manipulation of soliton ensembles by spectral filters

It was shown recently that an ensemble of solitons can be created in a driven optical fiber resonator. The ensemble can exist in either an ordered or a disordered state; these have been referred to as soliton solid or gas (fluid), respectively. We demonstrate through numerical simulation that the transition from one state to the other or vice versa, and further manipulation of the ensemble, can be effected through the insertion of spectral filters. Received: 7 July 2000 / Revised version: 1 September 2000 / Published online: 30 November 2000     

The Curious Nonexistence of Gaussian 2-Designs

Ensembles of pure quantum states whose 2nd moments equal those of the unitarily uniform Haar ensemble—2-designs—are optimal solutions for several tasks in quantum information science, especially state and process tomography. We show that Gaussian states cannot form a 2-design for the continuous-variable (quantum optical) Hilbert space ${L^2(\mathbb{R})}$ . This is surprising because the affine symplectic group HWSp (the natural symmetry group of Gaussian states) is irreducible on the symmetric subspace of two copies. In finite dimensional Hilbert spaces, irreducibility guarantees that HWSp-covariant ensembles (such as mutually unbiased bases in prime dimensions) are always 2-designs. This property is violated by continuous variables for a subtle reason: the (well-defined) HWSp-invariant ensemble of Gaussian states does not have a density matrix because its defining integral does not converge. In fact, no Gaussian ensemble is even close (in a precise sense) to being a 2-design. This surprising difference between discrete and continuous quantum mechanics has important implications for optical state and process tomography.     

Using simple particle shapes to model the Stokes scattering matrix of ensembles of wavelength-sized particles with complex shapes: possibilities and limitations

We investigate to what extent the full Stokes scattering matrix of an ensemble of wavelength-sized particles with complex shapes can be modeled by employing an ensemble of simple model shapes, such as spheres, spheroids, and circular cylinders. We also examine to what extent such a simple-shape particle model can be used to retrieve meaningful shape information about the complex-shaped particle ensemble. More specifically, we compute the Stokes scattering matrix for ensembles of randomly oriented particles having several polyhedral prism geometries of different sizes and shape parameters. These ensembles serve as proxies for size-shape mixtures of particles containing several different shapes of higher geometrical complexity than the simple-shaped model particles we employ. We find that the phase function of the complex-shaped particle ensemble can be accurately modeled with a size distribution of volume-equivalent spheres. The diagonal elements of the scattering matrix are accurately reproduced with a size-shape mixture of spheroids. A model based on circular cylinders accurately fits the full scattering matrix including the off-diagonal elements. However, the modeling results provide us with only a rough estimate of the effective shape parameter of the complex-shaped particle ensemble to be modeled. They do not allow us to infer detailed information about the shape distribution of the complex-shaped particle ensemble.     

State and parameter estimation in stochastic dynamical models

This paper derives generalized maximum likelihood estimates of state and model parameters of a stochastic dynamical model. In contrast to previous studies, the change in background distribution due to changes in model parameters is taken into account. An ensemble approach to solving the maximum likelihood estimates is proposed. An exact solution for the ensemble update based on a square root Kalman Filter is derived. This solution involves a two step procedure in which an ensemble is first produced by a standard ensemble Kalman Filter, and then “corrected” to account for parameter estimation, thereby allowing a user to take advantage of an existing ensemble filter. The solution is illustrated with simple, low-dimensional stochastic dynamical models and shown to work well and outperform augmentation methods for estimating stochastic parameters.     

Demonstrating various quantum effects with two entangled laser beams

We report on the preparation of entangled two mode squeezed states of yet unseen quality. Based on a measurement of the covariance matrix we found a violation of the Reid and Drummond EPR-criterion at a value of only 0.36 ± 0.03 compared to the threshold of 1. Furthermore, quantum state tomography was used to extract a single photon Fock state solely based on homodyne detection, demonstrating the strong quantum features of this pair of laser-beams. The probability for a single photon in this ensemble measurement exceeded 2/3.