A Method for the Time-Varying Nonlinear Prediction of Complex Nonstationary Biomedical Signals

A method to perform time-varying (TV) nonlinear prediction of biomedical signals in the presence of nonstationarity is presented in this paper. The method is based on identification of TV autoregressive models through expansion of the TV coefficients onto a set of basis functions and on k -nearest neighbor local linear approximation to perform nonlinear prediction. The approach provides reasonable nonlinear prediction even for TV deterministic chaotic signals, which has been a daunting task to date. Moreover, the method is used in conjunction with a TV surrogate method to provide statistical validation that the presence of nonlinearity is not due to nonstationarity itself. The approach is tested on simulated linear and nonlinear signals reproducing both time-invariant (TIV) and TV dynamics to assess its ability to quantify TIV and TV degrees of predictability and detect nonlinearity. Applicative examples relevant to heart rate variability and EEG analyses are then illustrated.     

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.     

Representation of time-varying nonlinear systems with time-varying principal dynamic modes

System identification of nonlinear time-varying (TV) systems has been a daunting task, as the number of parameters required for accurate identification is often larger than the number of data points available, and scales with the number of data points. Further, a 3-D graphical representation of TV second-order nonlinear dynamics without resorting to taking slices along one of the four axes has been a significant challenge to date. In this paper, we present a TV principal dynamic mode (TVPDM) method which overcomes these deficiencies. The TVPDM, by design, reduces one dimension, and by projecting PDM coefficients onto a set of basis functions, both nonstationary and nonlinear dynamics can be characterized. Another significant advantage of the TVPDM is its ability to discriminate the signal from noise dynamics, and provided that signal dynamics are orthogonal to each other, it has the capability to separate them. The efficacy of the proposed method is demonstrated with computer simulation examples comprised of various forms of nonstationarity and nonlinearity. The application of the TVPDM to the human heart rate and arterial blood pressure data during different postures is also presented and the results reveal significant nonstationarity even for short-term data recordings. The newly developed method has the potential to be a very useful tool for characterizing nonlinear TV systems, which has been a significant, challenging problem to date.     

Surrogate data analysis for assessing the significance of the coherence function

In cardiovascular variability analysis, the significance of the coupling between two time series is commonly assessed by setting a threshold level in the coherence function. While traditionally used statistical tests consider only the parameters of the adopted estimator, the required zero-coherence level may be affected by some features of the observed series. In this study, three procedures, based on the generation of surrogate series sharing given properties with the original but being structurally uncoupled, were considered: independent identically distributed (IID), Fourier transform (FT), and autoregressive (AR). IID surrogates maintained the distribution of the original series, while FT and AR surrogates preserved the power spectrum. The ability of the three methods to define the threshold for zero coherence was validated and compared by computer simulations reproducing typical cardiovascular interactions. While the IID threshold depended only on record length and design parameters of the coherence estimator, FT and AR thresholds were frequency-dependent with peaks corresponding to the local maxima of the estimated coherence. FT and AR surrogates were able to compensate spurious coherence peaks due to equal-frequency but independent oscillations in the two series. The benefit of frequency-dependent thresholds was evident for short series with narrow-band oscillations. Thus, surrogates preserving the power spectrum of the original series are recommended to avoid false coupling detections in the presence of oscillations occurring at nearby frequencies but produced by different mechanisms, as may frequently happen in cardiovascular and cardiorespiratory regulation.     

Application of linear and nonlinear time series modeling to heartrate dynamics analysis

The linear autoregressive (AR) model is often used to investigate the pathophysiologic mechanisms controlling heart rate (HR) dynamics. This study implemented parametric models new to this field to determine if a more appropriate HR dynamics modeling structure exists. The linear AR and autoregressive-moving average (ARMA) models, and the nonlinear polynomial autoregressive (PAR) and bilinear (BL) models were fit to instantaneous HR time series obtained from nine subjects in the supine position. Model orders were determined by the Akaike Information Criteria (AIC). Model residual variance was used as the primary intermodel comparison criterion, with significance evaluated by a λ² distributed statistic. The BL model best represented the HR dynamics, as its residual variance was significantly (p<0.05) smaller than that of the corresponding AR model for nine out of nine data sets. In all cases, the BL model had a smaller residual variance than either the ARMA or PAR models. The bilinear model was ineffective at data forecasting, however, the authors show that this cannot reflect BL model validity because poor prediction is inherent to the BL model structure. The apparent superiority of the nonlinear bilinear model suggests that future heart rate dynamics studies should put greater emphasis on nonlinear analyses     

一种基于KS检验的时间序列非线性检验方法

检验统计量的选取将对时间序列非线性检验的结果产生重要影响。该文在采用打乱相位法产生替代数据后,引入了一种非参数检验Kolmogorov-Smirnov 检验(简称KS检验)作为检验统计量。通过对各类信号的数值实验及与传统使用的高阶自相关量以及时间反演不可逆量对比结果表明,KS检验是一种有效、稳定的非线性检验统计量,对噪声信号具有较强的抗噪能力,而对非线性信号具有较高的敏感性。     

Time-varying autoregressive model-based multiple modes particle filtering algorithm for respiratory rate extraction from pulse oximeter

We present a particle filtering algorithm, which combines both time-invariant (TIV) and time-varying autoregressive (TVAR) models for accurate extraction of breathing frequencies (BFs) that vary either slowly or suddenly. The algorithm sustains its robustness for up to 90 breaths/min (b/m) as well. The proposed algorithm automatically detects stationary and nonstationary breathing dynamics in order to use the appropriate TIV or TVAR algorithm and then uses a particle filter to extract accurate respiratory rates from as low as 6 b/m to as high as 90 b/m. The results were verified on 18 healthy human subjects (16 for metronome and 2 for spontaneous measurements), and the algorithm remained accurate even when the respiratory rate suddenly changed by 24 b/m (either increased or decreased by this amount). Furthermore, simulation examples show that the proposed algorithm remains accurate for SNR ratios as low as -20 dB. We are not aware of any other algorithms that are able to provide accurate TV BF over a wide range of respiratory rates directly from pulse oximeters.     

Prediction of short cardiovascular variability signals based on conditional distribution

A new approach measuring the predictability of a process is proposed. The predictor is defined as the median of the distribution conditioned by a sequence of L - 1 previous samples (i.e., a pattern). A function referred to as the corrected mean squared predictor error is defined to prevent the perfect adequacy to the data (i.e., the decrease to zero of the prediction error), thus avoiding to divide the whole set of data in learning and test sets. This function exhibits a minimum and this minimum is taken as a measure of predictability of the series. The use of the minimization procedure avoids to fix a priori the pattern length L. This approach permits one a reliable measure of predictability on short data sequences (around 300 samples). Moreover, this method, in connection with a surrogate data approach, is useful to detect nonlinear dynamics. The analysis indicates that, in simulated and real data, predictability and nonlinearity measures provide different information. The application of this approach to the analysis of cardiovascular variability series of the heart period (RR interval) and systolic arterial pressure (SAP) shows: 1) SAP series is more predictable than RR interval series; 2) predictability of the RR interval series is larger during tilt, during controlled respiration at 10 breaths/min (bpm) and after high-dose administration of atropine; 3) SAP series is dominated by linear correlation; 4) RR interval series exhibits nonlinear dynamics during controlled respiration at 10 bpm and after low-dose administration of atropine, while it is linear during sympathetic activation produced by tilt and after peripheral parasympathetic blockade caused by high-dose administration of atropine.     

Complexity and nonlinearity in short-term heart period variability: comparison of methods based on local nonlinear prediction

This paper evaluates the paradigm that proposes to quantify short-term complexity and detect nonlinear dynamics by exploiting local nonlinear prediction. Local nonlinear prediction methods are classified according to how they judge similarity among patterns of L samples (i.e., according to different definitions of the cells utilized to discretize the phase space) and examined in connection with different types of surrogate data: 1) phase-randomized or Fourier transform based, FT; 2) amplitude-adjusted FT, AAFT; 3) iteratively-refined AAFT, IAAFT, preserving distribution IAAFT-1; 4) IAAFT preserving power spectrum, IAAFT-2. The methods were applied on ad-hoc simulations and on a large database of short heart period variability series (approximately 300 cardiac beats) recorded in healthy young subjects during experimental conditions inducing a sympathetic activation (head-up tilt, infusion of nitroprusside, or handgrip), a parasympathetic activation (low dose administration of atropine or infusion of phenylephrine), a complete parasympathetic blockade (high dose administration of atropine), or during controlled respiration at different breathing rates. As to complexity analysis we found that: 1) although complexity indexes derived from different methods were different in terms of absolute values, changes due to experimental conditions were consistently detected; 2) complexity was significantly decreased by all the experimental conditions provoking a sympathetic activation and by controlled respiration at slow breathing rates. As to detection of nonlinearities we found that: 1) IAAFT-1 and IAAFT-2 surrogates performed similarly in all protocols; 2) FT and IAAFT surrogates detected about the same percentage of nonlinear dynamics in all protocols; 3) AAFT surrogates were inappropriate with all the methods and should be dismissed in future applications; 4) methods based on overlapping cells with variable size were characterized by a larger rate of false detections of nonlinear dynamics; 5) short-term heart period variability at rest was mostly linear; 6) controlled respiration at slow breathing rates increased nonlinear components, while the separate activation of the two branches of the autonomic nervous system (i.e., sympathetic or parasympathetic) was ineffective at this regard.     

Nonlinear considerations in EEG signal classification

We investigate the effect of incorporating modeling of nonlinearity on the classification of electroencephalogram (EEG) signals using an artificial neural network (ANN). It is observed that the ANN's predictive ability is improved after preprocessing EEG signals using a particular nonlinear modeling technique, viz. a bilinear model, compared with those obtained by using a particular classical linear analysis method, viz. an autoregressive (AR) model. Until recently, linear time-invariant Gaussian modeling has dominated the development of time series modeling and feature extraction. The advantage of such classical models lies in the fact that a complete signal processing theory is available. In the case of EEG signals, where the underlying theory regarding the dynamical law governing the generation of these signals (e,g., the underlying physiological factors) is not completely understood, a case can be made for using improved signal processing models that are not subject to linear constraints. Such models should recognize important features of the observed data that may not be well modeled by a linear time-invariant model. It is known that EEG signals are nonstationary, and it is possible that they may be nonlinear as well. Thus, one way of gaining further insights on the structure of EEG signals is to introduce nonlinear models and higher order spectra. This paper compares the results of classification using a linear AR model with those obtained from a bilinear model. It is shown that in certain cases, the nonlinearity of the EEG signals is an important factor that ought to be taken into consideration during preprocessing of the signals prior to the classification task     

一种定量检验多维信号非线性的方法

利用多变量时间序列替代数据生成原理生成了实测多变量时间序列的多组替代时间序列,由广义关联积分计算了实测时间序列和替代时间序列的广义冗余,提出了一种用线性冗余和广义冗余作为显著性检验统计量定量检验多维信号非线性的方法。一个线性自回归模型和Lorenz系统的仿真计算验证了这种方法的有效性。     

Orthonormal-basis partitioning and time-frequency representation of cardiac rhythm dynamics

Although a number of time-frequency representations have been proposed for the estimation of time-dependent spectra, the time-frequency analysis of multicomponent physiological signals, such as beat-to-beat variations of cardiac rhythm or heart rate variability (HRV), is difficult. We thus propose a simple method for 1) detecting both abrupt and slow changes in the structure of the HRV signal, 2) segmenting the nonstationary signal into the less nonstationary portions, and 3) exposing characteristic patterns of the changes in the time-frequency plane. The method, referred to as orthonormal-basis partitioning and time-frequency representation (OPTR), is validated using simulated signals and actual HRV data. Here we show that OPTR can be applied to long multicomponent ambulatory signals to obtain the signal representation along with its time-varying spectrum.     

Equalization of rapidly fading channels: self-recovering methods

Blind estimation of time-varying (TV), rapidly fading channels is addressed, using a basis expansion approach. Each TV coefficient is expanded onto a basis and the expansion parameters are estimated for subsequent use in Viterbi or decision-feedback equalizers. Blind estimation of the expansion parameters is accomplished using higher order statistics. Identifiability of the channel is shown from second and fourth-order TV cumulants of the received signal. A cumulant matching criterion is adopted and instantaneous approximations are proposed in place of the nonstationary ensembles. Linear methods are also derived to initialize the nonlinear optimization procedure. Strong convergence of the proposed method is established. Finally, the method is tested on a simulated mobile radio channel with multipath     

Time-Frequency ARMA Models and Parameter Estimators for Underspread Nonstationary Random Processes

Parsimonious parametric models for nonstationary random processes are useful in many applications. Here, we consider a nonstationary extension of the classical autoregressive moving-average (ARMA) model that we term the time-frequency autoregressive moving-average (TFARMA) model. This model uses frequency shifts in addition to time shifts (delays) for modeling nonstationary process dynamics. The TFARMA model and its special cases, the TFAR and TFMA models, are shown to be specific types of time-varying ARMA (AR, MA) models. They are attractive because of their parsimony for underspread processes, that is, nonstationary processes with a limited time-frequency correlation structure. We develop computationally efficient order-recursive estimators for the TFARMA, TFAR, and TFMA model parameters which are based on linear time-frequency Yule-Walker equations or on a new time-frequency cepstrum. Simulation results demonstrate that the proposed parameter estimators outperform existing estimators for time-varying ARMA (AR, MA) models with respect to accuracy and/or numerical efficiency. An application to the time-varying spectral analysis of a natural signal is also discussed.     

Bayesian approach to parameter estimation and interpolation oftime-varying autoregressive processes using the Gibbs sampler

A nonstationary time series is one in which the statistics of the process are a function of time; this time dependency makes it impossible to utilise standard analytically defined statistical estimators to parameterise the process. To overcome this difficulty, the time series is considered within a finite time interval and is modelled as a time-varying autoregressive (AR) process. The AR coefficients that characterise this process are functions of time, represented by a family of basis vectors. The corresponding basis coefficients are invariant over the time window and have stationary statistical properties. A method is described for applying a Markov chain Monte Carlo method known as the Gibbs sampler to the problem of estimating the parameters of such a time-varying autoregressive (TVAR) model, whose time dependent coefficients are modelled by basis functions. The Gibbs sampling scheme is then extended to include a stage which may be used for interpolation. Results on synthetic and real audio signals show that the model is flexible, and that a Gibbs sampling framework is a reasonable scheme for estimating and characterising a time-varying AR process     

Background millimeter radiation influence in cardiology on patients with metabolic and pre-metabolic syndrome

The effects of background millimeter radiations (BMR) in patients with coronary artery disease (CAD), hypertension and in subjects with Inherited real risk of CAD, were investigated through invariant statistic measures, typical of nonlinear dynamics analysis of biological systems. The experimental evidences show that BMR ameliorate the nonlinear complexity in biosystems, recognized sign of physiological behavior, by increasing both the rate of unpredictability of heart rate variability (HRV) in patients with metabolic syndrome and the fractal dimension of coronary microvessel oscillations in subjects with pre-metabolic syndrome, healing their genetic alteration and CAD Inherited real risk.     

Generalized feature extraction for time-varying autoregressivemodels

In this paper, a feature extraction scheme for a general type of nonstationary time series is described. A non-stationary time series is one in which the statistics of the process are a function of time; this time dependency makes it impossible to utilize standard globally derived statistical attributes such as autocorrelations, partial correlations, and higher order moments as features. In order to overcome this difficulty, the time series vectors are considered within a finite-time interval and are modeled as time-varying autoregressive (AR) processes. The AR coefficients that characterize the process are functions of time that may be represented by a family of basis vectors. A novel Bayesian formulation is developed that allows the model order of a time-varying AR process as well as the form of the family of basis vectors used in the representation of each of the AR coefficients to be determined. The corresponding basis coefficients are then invariant over the time window and, since they directly relate to the time-varying AR coefficients, are suitable features for discrimination. Results illustrate the effectiveness of the method     

Time-variant power spectrum analysis for the detection of transientepisodes in HRV signal

A time-variant algorithm of autoregressive (AR) identification is introduced and applied to the heart rate variability (HRV) signal. The power spectrum is calculated from the AR coefficients derived from each single RR interval considered. Time-variant AR coefficients are determined through adaptive parametric identification with a forgetting factor which obtains weighted values on a running temporal window of 50 preceding measurements. Power spectrum density (PSD) is hence obtained at each cardiac cycle, making it possible to follow the dynamics of the spectral parameters on a beat-by-beat basis. These parameters are mainly the LF (low-frequency) and the HF (high-frequency) powers, and their ratio, LF/HF. These together account for the balanced sympatho-vagal control mechanism affecting the heart rate. This method is applied to subjects suffering from transient ischemic attacks. The time variant spectral parameters suggest an early activation of LF component in the HRV power spectrum. It precedes by approximately 1.5-2 min the tachycardia and the ST displacement, generally indicative of the onset of an ischemic episode     

Recurrent statistics of nonstationary time series

Basic statistics of a nonstationary time series are estimated from its single realization. The estimates are represented in the form of a recurrent procedure forming residual time series and smoothing them with the help of effective models of digital data filtering or a locally weighted polynomial regression. The concept of local time weighting and robust weighting of residual series is generalized on the basis of a rational combination of models of distance-weighted least squares and an exponentially weighted regression. The estimation of trends, volatility, and autocorrelation for time series of companies’ sales volumes and the price dynamics of stock assets is simulated, and the results of simulation are presented.