Time-Varying Surrogate Data to Assess Nonlinearity in Nonstationary Time Series: Application to Heart Rate Variability

We propose a method to extend to time-varying (TV) systems the procedure for generating typical surrogate time series, in order to test the presence of nonlinear dynamics in potentially nonstationary signals. The method is based on fitting a TV autoregressive (AR) model to the original series and then regressing the model coefficients with random replacements of the model residuals to generate TV AR surrogate series. The proposed surrogate series were used in combination with a TV sample entropy (SE) discriminating statistic to assess nonlinearity in both simulated and experimental time series, in comparison with traditional time-invariant (TIV) surrogates combined with the TIV SE discriminating statistic. Analysis of simulated time series showed that using TIV surrogates, linear nonstationary time series may be erroneously regarded as nonlinear and weak TV nonlinearities may remain unrevealed, while the use of TV AR surrogates markedly increases the probability of a correct interpretation. Application to short (500 beats) heart rate variability (HRV) time series recorded at rest (R), after head-up tilt (T), and during paced breathing (PB) showed: (1) modifications of the SE statistic that were well interpretable with the known cardiovascular physiology; (2) significant contribution of nonlinear dynamics to HRV in all conditions, with significant increase during PB at 0.2 Hz respiration rate; and (3) a disagreement between TV AR surrogates and TIV surrogates in about a quarter of the series, suggesting that nonstationarity may affect HRV recordings and bias the outcome of the traditional surrogate-based nonlinearity test.     

A new heart rate variability analysis method by means of quantifying the variation of nonlinear dynamic patterns

A new heart rate variability (HRV) analysis method, quantifying the variation of nonlinear dynamic pattern (VNDP) in heart rate series, is proposed and validated against the age stratified Fantasia database. The method is based on three processes: (1) a recurrence quantification analysis (RQA) to quantify the dynamic patterns, (2) the use of mutual information (MI) and the entropy (EN) to characterize the VNDP, and 3) linear discriminant analysis to exploit the associations within MI and EN measures. Practically, the VNDP method overcomes the nonstationarity problem and exploits the nonstationary properties in HRV analyses. Physiologically, the VNDP reflects the properties of the fundamental short-term HRV dynamic system and the external associations of the system within the autonomous nervous system (ANS). The characteristic probability density peaks portrayed by VNDP plots indicate the quantum-like heart dynamics, which may provide valuable insights into the control of the ANS. The discrimination results of the reduced pattern dynamic range due to aging, from a new perspective, display the reduction in HRV. The significantly improved discriminatory power, compared to conventional RQA analyses, shows that the VNDP analysis can practically quantify the nonstationary nonlinear dynamics for ANS assessments.     

改进的高效动态时间规整算法语音识别系统

动态时间规整算法是结合了动态时间规整(DTW)技术和距离测度计算技术的一种非线性规整算法,在语音识别模板匹配中有重要的应用。为此提出一种改进的高效动态时间规整算法,其能有效加快搜索路径的寻找。基于Matlab实现了隐马尔科夫算法、高效动态时间规整算法和改进的高效动态时间规整算法的语音识别系统,同时进行了算法的仿真实验。实验结果表明,基于改进高效动态时间规整算法的训练速度远大于基于隐马尔可夫算法和高效动态时间规整算法的训练速度,而识别率下降很小,对于小词汇量非连续语音识别中高效动态时间规整算法的识别率为97.56%,隐马尔可夫算法的识别率为97.14%,改进高效动态时间规整算法的识别率为96.43%。     

Local holder exponent analysis of heart rate variability in preterm infants

Heart rate variability (HRV) displays scale-invariant fractal properties. Recent studies have revealed multifractal properties in the healthy human HRV, which could be characterized by singularities with various strength of local H?lder exponents embedded in HRV. In this paper, HRV time series from preterm infants, whose autonomic nervous system undergoes dramatic development, were collected longitudinally. Changes in fractality/multifractality of those HRV time series as the postmenstrual age were examined in order to see if they could quantify development of the autonomic nervous system. Temporal structure of the singularities at several representative time scales was also analyzed to show that intersingular event intervals could be well described by "power law distribution," and the singular events appeared with age-dependent long-range correlation in its strength. Detailed analyses suggested that fractality and multifractality of HRV, respectively, could quantify the development of the respiratory center and the parasympathetic nervous system in the preterm infants. The results obtained in this study might be beneficial for detecting occurrences of life threatening singular events such as big apnea in preterm infants.     

Constituent factors of heart rate variability ALLSTAR big data analysis

Various indices have been reported regarding heart rate variability (HRV), but many of them correlate to each other, suggesting the existence of the underlying common factors. We tried to extract factors underlying HRV indices and investigated their features. Using big data of 24-h electrocardiogram (ECG) called Allostatic State Mapping by Ambulatory ECG Repository (ALLSTAR), we calculated 4 time-domain, 4 frequency-domain, and 2 nonlinear HRV indices and the amplitude of cyclic variation of heart rate (Acv) in 113,793 men and 140,601 women with sinus rhythm ECG. Factor analysis revealed that there were two factors with eigenvalue?≥?1 by which 91% of variance among the HRV indices was explained. Factor 1 that was strongly contributed by very-low frequency, low frequency (LF), and high frequency (HF) components and Acv and it increased with age from 0 to 20 year, then decreased until 65 year, and increased slightly after 80 year. It also increased with daily physical activity at the mild level of activity. Factor 2 that was contributed strongly by scaling exponent α₁ and LF-to-HF ratio increased with age until 35 year, plateaued between 35 and 55 year, and decreased thereafter. It also increased with mild to moderate physical activity. HRV indices are constituted by two common factors relating to cardiac vagal function and complexity of heart rate dynamics, respectively, which differ in the relationships with age and physical activity from each other. Although many indices have been proposed for HRV, their constituent factors may be a few.

     

针对时间序列多步预测的聚类隐马尔科夫模型

时间序列的预测在现今社会各个领域中有着广泛的应用.本文针对时间序列趋势预测中的多步预测问题,提出了基于聚类的隐马尔科夫模型,利用隐马尔科夫模型中的隐状态来表示产生时间序列数据时的系统内部状态,实现对多步时间序列的预测.针对时间序列聚类中的距离计算问题,提出结合时间序列时间性和相似性的聚类算法,并给出了迭代精化基于聚类的隐马尔科夫模型的方法.实验表明,本文提出的方法在时间序列多步预测中精度较高.     

Dynamic system identification with order statistics

Systems consisting of linear dynamic and memory-less nonlinear subsystems are identified. The paper deals with systems in which the nonlinear element is followed by a linear element, as well as systems in which the subsystems are connected in parallel. The goal of the identification is to recover the nonlinearity from noisy input-output observations of the whole system; signals interconnecting the elements are not measured. Observed values of the input signal are rearranged in increasing order, and coefficients for the expansion of the nonlinearity in trigonometric series are estimated from the new sequence of observations obtained in this way. Two algorithms are presented, and their mean integrated square error is examined. Conditions for pointwise convergence are also established. For the nonlinearity satisfying the Lipschitz condition, the error converges to zero. The rate of convergence derived for differentiable nonlinear characteristics is insensitive to the roughness of the probability density of the input signal. Results of numerical simulation are also presented     

Application of extreme value theory to level estimation innonlinearly distorted hidden Markov models

This paper is concerned with the application of extreme value theory (EVT) to the state level estimation problem for discrete-time, finite-state Markov chains hidden in additive colored noise and subjected to unknown nonlinear distortion. If the nonlinear distortion affects only those observations with small magnitudes or those that lie outside a finite interval, we show that the level estimation problem can be reduced to a curve fitting problem with a unique global minimum. Compared with optimum maximum likelihood estimation algorithms, the developed level estimation algorithms are computationally inexpensive and are not affected by the unknown nonlinearity as long as the extreme values of observations are not distorted. This work has been motivated by unknown deadzone and saturation nonlinearities introduced by sensors in data measurement systems. We illustrate the effectiveness of the new EVT-based level estimation algorithms with computer simulations     

Analyticity of Entropy Rate of Hidden Markov Chains

We prove that under mild positivity assumptions the entropy rate of a hidden Markov chain varies analytically as a function of the underlying Markov chain parameters. A general principle to determine the domain of analyticity is stated. An example is given to estimate the radius of convergence for the entropy rate. We then show that the positivity assumptions can be relaxed, and examples are given for the relaxed conditions. We study a special class of hidden Markov chains in more detail: binary hidden Markov chains with an unambiguous symbol, and we give necessary and sufficient conditions for analyticity of the entropy rate for this case. Finally, we show that under the positivity assumptions, the hidden Markov chain itself varies analytically, in a strong sense, as a function of the underlying Markov chain parameters     

Individual time-dependent spectral boundaries for improved accuracy in time-frequency analysis of heart rate variability

Heart rate variability (HRV) is a major noninvasive technique for evaluating the autonomic nervous system (ANS). Use of time-frequency approach to analyze HRV allows investigating the ANS behavior from the power integrals, as a function of time, in both steady-state and non steady-state. Power integrals are examined mainly in the low-frequency and the high-frequency bands. Traditionally, constant boundaries are chosen to determine the frequency bands of interest. However, these ranges are individual, and can be strongly affected by physiologic conditions (body position, breathing frequency). In order to determine the dynamic boundaries of the frequency bands more accurately, especially during autonomic challenges, we developed an algorithm for the detection of individual time-dependent spectral boundaries (ITSB). The ITSB was tested on recordings from a series of standard autonomic maneuvers with rest periods between them, and the response to stand was compared to the known physiological response. A major advantage of the ITSB is the ability to reliably define the mid-frequency range, which provides the potential to investigate the physiologic importance of this range.     

Dynamic Quantizer Design for Hidden Markov State Estimation Via Multiple Sensors With Fusion Center Feedback

This paper considers the state estimation of hidden Markov models by sensor networks. The objective is to minimize the long term average of the mean square estimation error for the underlying finite state Markov chain. By employing feedback from the fusion center, a dynamic quantization scheme for the sensor nodes is proposed and analyzed by a stochastic control approach. Dynamic rate allocation is also considered when the sensor nodes generate mode dependent measurements.     

一种基于加权隐马尔可夫的 自回归状态预测模型

针对电子系统状态趋势预测问题,提出了一种加权隐马尔可夫模型的自回归趋势预测方法.该方法以自回归模型作为隐马尔可夫的状态输出,利用加权预测思想对马尔可夫链中的隐状态进行混合高斯模型的加权序列预测,并利用最大概率隐状态下的自回归系数计算模型输出.通过对实际的复杂混沌序列和电子系统BIT状态数据进行趋势预测,并针对不同模型参数下的预测结果进行实验分析,结果表明该方法对系统状态变化的趋势具有较好的预测性能.     

基于小波域隐马尔可夫混合模型的SAR图像降斑算法

在小波域马尔可夫随机场(MRF)和隐马尔可夫树(HMT)的基础上,提出了一种新的合成孔径雷达(SAR)图像降斑算法.该算法在对乘性噪声不取对数变换的情况下,融合了贝叶斯最小均方误差(MMSE)抑制噪声技术.为了提高HMT的速度,采用了一个新的隐马尔可夫半树模型,该模型考虑了小波系数的持续性和聚类性,分别用HMT和MRF刻画.仿真结果表明该算法在抑制斑点噪声的同时,有效的保持了边缘,避免对数变换带来的一些误差,取得了好的效果,其速度比HMT模型提高了二十倍.     

Two-Time-Scale Approximation for Wonham Filters

This paper is concerned with approximation of Wonham filters. A focal point is that the underlying hidden Markov chain has a large state space. To reduce computational complexity, a two-time-scale approach is developed. Under time scale separation, the state space of the underlying Markov chain is divided into a number of groups such that the chain jumps rapidly within each group and switches occasionally from one group to another. Such structure gives rise to a limit Wonham filter that preserves the main features of the filtering process, but has a much smaller dimension and therefore is easier to compute. Using the limit filter enables us to develop efficient approximations and useful filters for hidden Markov chains. The main advantage of our approach is the reduction of dimensionality     

基于稀疏贝叶斯学习的可见光通信系统中LED非线性补偿技术

针对可见光通信(visible light communication,VLC)系统中LED非线性引起信号失真的问题,提出一种基于稀疏贝叶斯学习的接收端LED非线性补偿技术.该技术利用稀疏贝叶斯框架下的相关向量机(relevance vector machine,RVM)回归模型对LED的非线性传输特性进行估计,从而在接收端对LED非线性引起的信号失真进行时域补偿.仿真实验表明,与常用的沃尔泰拉(Volterra)非线性补偿技术相比,该技术能够更加有效地抑制LED非线性引起的信号失真,从而可以更加充分利用LED的动态范围来增大系统容量.     

Nonlinear analysis of the separate contributions of autonomic nervous systems to heart rate variability using principal dynamic modes

This paper introduces a modified principal dynamic modes (PDM) method, which is able to separate the dynamics of sympathetic and parasympathetic nervous activities. The PDM is based on the principle that among all possible choices of expansion bases, there are some that require the minimum number of basis functions to achieve a given mean-square approximation of the system output. Such a minimum set of basis functions is termed PDMs of the nonlinear system. We found that the first two dominant PDMs have similar frequency characteristics for parasympathetic and sympathetic activities, as reported in the literature. These results are consistent for all nine of our healthy human subjects using our modified PDM approach. Validation of the purported separation of parasympathetic and sympathetic activities was performed by the application of the autonomic nervous system blocking drugs atropine and propranolol. With separate applications of the respective drugs, we found a significant decrease in the amplitude of the waveforms that correspond to each nervous activity. Furthermore, we observed near complete elimination of these dynamics when both drugs were given to the subjects. Comparison of our method to the conventional low-frequency/high-frequency ratio shows that our proposed approach provides more accurate assessment of the autonomic nervous balance. Our nonlinear PDM approach allows a clear separation of the two autonomic nervous activities, the lack of which has been the main reason why heart rate variability analysis has not had wide clinical acceptance.     

State-dependent vector hybrid linear and nonlinear ARMA modeling: Theory

A new model is proposed to represent a general vector nonstationary and nonlinear process by setting up a state-dependent vector hybrid linear and nonlinear autoregressive moving average (SVH-ARMA) model. The linear part of the process is represented by a vector ARMA model, the nonlinear part is represented by a vector nonlinear ARMA model employing a multilayer feedforward neural network, and the nonstationary characteristics are captured with a hidden Markov chain. Based on a unifiedQ-likelihood function, an expectation-maximization algorithm for model identification is derived, and the model parameters are estimated by applying a state-dependent training and nonlinear optimization technique iteratively, which finally yields maximum likelihood estimation of model parameters. This model can track the nonstationary varying of a vector linear and/or nonlinear process adaptively and represent a vector linear and/or nonlinear system with low order. Moreover, it is able to characterize and track the long-range, second-order correlation features of many time series and thus can be used for reliable multiple step ahead prediction. Some impressive applications of the SVH-ARMA model are being presented in the companion paper by Zheng et al., pp. 575–597, this issue.     

State-dependent vector hybrid linear and nonlinear ARMA modeling: Applications

In a recent companion paper, a new method has been presented for modeling general vector nonstationary and nonlinear processes based on a state-dependent vector hybrid linear and nonlinear autoregressive moving average (SVH-ARMA) model. This paper discusses some potential applications of the SVH-ARMA model, including signal filtering, time series prediction, and system control. First, a state-space model governed by a hidden Markov Chain is shown to be equivalent to the SVH-ARMA model. Based on this state-space model, the extended Kalman filtering and Bayesian estimation techniques are applied for noisy signal enhancement. The result of a noisy image enhancement verifies that the model can track the time-varying statistical characteristics of nonstationary and nonlinear processes adaptively. Second, the SVH-ARMA model is used for a vector time series prediction, which can attain more accurate multiple step ahead prediction, than conventional forecasting methods. Third, a new technique is developed for predicting scalar long correlation time series in the wavelet scale space domain based on the SVH-ARMA model. Dyadic wavelet transform is employed to convert a scalar time series to a vector time series, to which the SVH-ARMA model is applied for vector time series prediction. More accurate and robust forecasting results in both one step and multiple step ahead prediction can be gained. See also the companion paper on theory, by Zheng et al., pp. 551–574, this issue.     

Optimal functional expansion for estimation from counting observations

A systematic method to develop approximate nonlinear estimators is presented, in the form of a functional series, for the signal that modulates the rate of a counting process. The estimators are optimal for the given structure and approach the minimum variance (MV) estimator when the approximation order increases. Two kinds of functional series, the iterated integral (II) series and the Fourier-Charlier (FC) series, are used. Product-to-sum formulas for the II and FC functionals are derived. By using the formulas, the MV estimate is projected onto the Hilbert subspaces of the II and the FC series driven by the counting observations with the given index set. The projection results in a Wiener-Hopf type equation for the II kernels and a system of linear algebraic equations for the FC coefficients. The FC series estimator consists of finitely many single Wiener integrals of the counting observations and a nonlinear postprocessor. The nonlinear postprocessor, however, is not memoryless.     

On adaptive HMM state estimation

New online adaptive hidden Markov model (HMM) state estimation schemes are developed, based on extended least squares (ELS) concepts and recursive prediction error (RPE) methods. The best of the new schemes exploit the idempotent nature of Markov chains and work with a least squares prediction error index, using a posterior estimates, more suited to Markov models than traditionally used in identification of linear systems. These new schemes learn the set of N Markov chain states, and the a posteriori probability of being in each of the states at each time instant. They are designed to achieve the strengths, in terms of computational effort and convergence rates, of each of the two classes of earlier proposed adaptive HMM schemes without the weaknesses of each in these areas. The computational effort is of order N. Implementation aspects of the proposed algorithms are discussed, and simulation studies are presented to illustrate convergence rates in comparison to earlier proposed online schemes