A new algorithm for linear and nonlinear ARMA model parameter estimation using affine geometry

Physiology-Informed Real-Time Mean Arterial Blood Pressure Learning and Prediction for Septic Patients Receiving Norepinephrine

OBJECTIVE

Septic shock is a life-threatening manifestation of infection with a mortality of 20-50% [1]. A catecholamine vasopressor, norepinephrine (NE), is widely used to treat septic shock primarily by increasing blood pressure. For this reason, future blood pressure knowledge is invaluable for properly controlling NE infusion rates in septic patients. However, recent machine learning and data-driven methods often treat the physiological effects of NE as a black box. In this paper, a real-time, physiology-informed human mean arterial blood pressure model for septic shock patients undergoing NE infusion is studied.

METHODS

Our methods combine learning theory, adaptive filter theory, and physiology. We learn least mean square adaptive filters to predict three physiological parameters (heart rate, pulse pressure, and the product of total arterial compliance and arterial resistance) from previous data and previous NE infusion rate. These predictions are combined according to a physiology model to predict future mean arterial blood pressure.

RESULTS

Our model successfully forecasts mean arterial blood pressure on 30 septic patients from two databases. Specifically, we predict mean arterial blood pressure 3.33 minutes to 20 minutes into the future with a root mean square error from 3.56 mmHg to 6.22 mmHg. Additionally, we compare the computational cost of different models and discover a correlation between learned NE response models and a patient's SOFA score.

CONCLUSION

Our approach advances our capability to predict the effects of changing NE infusion rates in septic patients.

SIGNIFICANCE

More accurately predicted MAP can lessen clinicians' workload and reduce error in NE titration.

Fuzzy neural network technique for system state forecasting

In many system state forecasting applications, the prediction is performed based on multiple datasets, each corresponding to a distinct system condition. The traditional methods dealing with multiple datasets (e.g., vector autoregressive moving average models and neural networks) have some shortcomings, such as limited modeling capability and opaque reasoning operations. To tackle these problems, a novel fuzzy neural network (FNN) is proposed in this paper to effectively extract information from multiple datasets, so as to improve forecasting accuracy. The proposed predictor consists of both autoregressive (AR) nodes modeling and nonlinear nodes modeling; AR models/nodes are used to capture the linear correlation of the datasets, and the nonlinear correlation of the datasets are modeled with nonlinear neuron nodes. A novel particle swarm technique [i.e., Laplace particle swarm (LPS) method] is proposed to facilitate parameters estimation of the predictor and improve modeling accuracy. The effectiveness of the developed FNN predictor and the associated LPS method is verified by a series of tests related to Mackey-Glass data forecast, exchange rate data prediction, and gear system prognosis. Test results show that the developed FNN predictor and the LPS method can capture the dynamics of multiple datasets effectively and track system characteristics accurately.

Dynamic assessment of baroreflex control of heart rate during induction of propofol anesthesia using a point process method

In this article, we present a point process method to assess dynamic baroreflex sensitivity (BRS) by estimating the baroreflex gain as focal component of a simplified closed-loop model of the cardiovascular system. Specifically, an inverse Gaussian probability distribution is used to model the heartbeat interval, whereas the instantaneous mean is identified by linear and bilinear bivariate regressions on both the previous R-R intervals (RR) and blood pressure (BP) beat-to-beat measures. The instantaneous baroreflex gain is estimated as the feedback branch of the loop with a point-process filter, while the RR-->BP feedforward transfer function representing heart contractility and vasculature effects is simultaneously estimated by a recursive least-squares filter. These two closed-loop gains provide a direct assessment of baroreflex control of heart rate (HR). In addition, the dynamic coherence, cross bispectrum, and their power ratio can also be estimated. All statistical indices provide a valuable quantitative assessment of the interaction between heartbeat dynamics and hemodynamics. To illustrate the application, we have applied the proposed point process model to experimental recordings from 11 healthy subjects in order to monitor cardiovascular regulation under propofol anesthesia. We present quantitative results during transient periods, as well as statistical analyses on steady-state epochs before and after propofol administration. Our findings validate the ability of the algorithm to provide a reliable and fast-tracking assessment of BRS, and show a clear overall reduction in baroreflex gain from the baseline period to the start of propofol anesthesia, confirming that instantaneous evaluation of arterial baroreflex control of HR may yield important implications in clinical practice, particularly during anesthesia and in postoperative care.

An autoregressive model-based particle filtering algorithms for extraction of respiratory rates as high as 90 breaths per minute from pulse oximeter

We present particle filtering (PF) algorithms for an accurate respiratory rate extraction from pulse oximeter recordings over a broad range: 12-90 breaths/min. These methods are based on an autoregressive (AR) model, where the aim is to find the pole angle with the highest magnitude as it corresponds to the respiratory rate. However, when SNR is low, the pole angle with the highest magnitude may not always lead to accurate estimation of the respiratory rate. To circumvent this limitation, we propose a probabilistic approach, using a sequential Monte Carlo method, named PF, which is combined with the optimal parameter search (OPS) criterion for an accurate AR model-based respiratory rate extraction. The PF technique has been widely adopted in many tracking applications, especially for nonlinear and/or non-Gaussian problems. We examine the performances of five different likelihood functions of the PF algorithm: the strongest neighbor, nearest neighbor (NN), weighted nearest neighbor (WNN), probability data association (PDA), and weighted probability data association (WPDA). The performance of these five combined OPS-PF algorithms was measured against a solely OPS-based AR algorithm for respiratory rate extraction from pulse oximeter recordings. The pulse oximeter data were collected from 33 healthy subjects with breathing rates ranging from 12 to 90 breaths/ min. It was found that significant improvement in accuracy can be achieved by employing particle filters, and that the combined OPS-PF employing either the NN or WNN likelihood function achieved the best results for all respiratory rates considered in this paper. The main advantage of the combined OPS-PF with either the NN or WNN likelihood function is that for the first time, respiratory rates as high as 90 breaths/min can be accurately extracted from pulse oximeter recordings.

Respiratory rate extraction via an autoregressive model using the optimal parameter search criterion

We present an autoregressive model-based method which enables accurate respiratory rate extraction from pulse oximeter recordings over a wide range: 12-48 breaths/min. The method uses the optimal parameter search (OPS) technique to estimate accurate AR parameters which are then factorized into multiple pole terms. The pole with the highest magnitude is shown to correspond to the respiratory rate. The performance of the proposed method to extract respiratory rate is compared to the widely used Burg algorithm using both simulation examples and pulse oximeter recordings. In a previous study, we demonstrated several nonparametric time-frequency approaches that were more accurate than Burg's algorithm when the data length was 1 min [Chon, K. H., S. Dash, and K. Ju. IEEE Trans. Biomed. Eng. 56(8):2054-2063, 2009]. One of the key advantages of the AR method is that a shorter data length can be used. Thus, in this study, we reduced the data length to 30 s and applied our OPS algorithm to examine if accurate respiratory rates can be extracted directly from pulse oximeter recordings. It was found that our proposed method's accuracy was consistently better with smaller variance than Burg's method. In particular, our proposed method's accuracy was significantly greater when respiratory rates were lower than 24 breaths/min.

Modelling and disentangling physiological mechanisms: linear and nonlinear identification techniques for analysis of cardiovascular regulation

Cardiovascular (CV) regulation is the result of a number of very complex control interactions. As computational power increases and new methods for collecting experimental data emerge, the potential for exploring these interactions through modelling increases as does the potential for clinical application of such models. Understanding these interactions requires the application of a diverse set of modelling techniques. Several recent mathematical modelling techniques will be described in this review paper. Starting from Granger's causality, the problem of closed-loop identification is recalled. The main aspects of linear identification and of grey-box modelling tailored to CV regulation analysis are summarized as well as basic concepts and trends for nonlinear extensions. Sensitivity analysis is presented and discussed as a potent tool for model validation and refinement. The integration of methods and models is fostered for a further physiological comprehension and for the development of more potent and robust diagnostic tools.

Parametric and nonparametric methods to generate time-varying surrogate data

We present both nonparametric and parametric approaches to generating time-varying surrogate data. Nonparametric and parametric approaches are based on the use of the short-time Fourier transform and a time-varying autoregressive model, respectively. Time-varying surrogate data (TVSD) can be used to determine the statistical significance of the linear and nonlinear coherence function estimates. Two advantages of the TVSD are that it keeps one from having to make an arbitrary decision about the significance of the coherence value, and it properly takes into account statistical significance levels, which may change with time. Our simulation examples and experimental results on blood pressure and heart rate data demonstrate the efficacy and applicability of the proposed TVSD methods.

Intracranial pressure waves: characterization of a pulsation absorber with notch filter properties using systems analysis: laboratory investigation

OBJECT

The relationship between the waveform of intracranial pressure (ICP) and arterial blood pressure can be quantitatively characterized using a newly developed technique in systems analysis, the time-varying transfer function. This technique considers the arterial blood pressure as an input signal composed of multiple frequencies represented in the output ICP according to the transfer function imposed by the intracranial system on the input signal. The transfer function can change with time and with physiological manipulations. The authors examined data obtained from canine experiments involving manipulations of ICP.

METHODS

The authors analyzed 11 experiments from 3 normal mongrel dogs under conditions of normal ICP and with changes in ICP made by bolus injection, infusion, or withdrawal of cerebrospinal fluid by using time-varying transfer function.

RESULTS

During normal ICP periods, the gain of the transfer function displayed a deep notch (> or = 1 log unit) centered at or near the cardiac frequency. In systems terms, the intracranial compartment under normal conditions appears to act as a notch filter attenuating the cardiac frequency input relative to other frequencies. Epochs of ICP elevation showed suppression of the notch, and the notch was restored when ICP returned to normal.

CONCLUSIONS

The intracranial system in these animals could be considered to include a pulsation absorber for which the target frequency appears to be close to the cardiac frequency. One possible source for such an absorber mechanism might be the free movement of cerebrospinal fluid, implying that impairment of this motion may have important clinical implications in various neurological conditions such as hydrocephalus.

Analysis of nonstationarity in renal autoregulation mechanisms using time-varying transfer and coherence functions

The extent to which renal blood flow dynamics vary in time and whether such variation contributes substantively to dynamic complexity have emerged as important questions. Data from Sprague-Dawley rats (SDR) and spontaneously hypertensive rats (SHR) were analyzed by time-varying transfer functions (TVTF) and time-varying coherence functions (TVCF). Both TVTF and TVCF allow quantification of nonstationarity in the frequency ranges associated with the autoregulatory mechanisms. TVTF analysis shows that autoregulatory gain in SDR and SHR varies in time and that SHR exhibit significantly more nonstationarity than SDR. TVTF gain in the frequency range associated with the myogenic mechanism was significantly higher in SDR than in SHR, but no statistical difference was found with tubuloglomerular (TGF) gain. Furthermore, TVCF analysis revealed that the coherence in both strains is significantly nonstationary and that low-frequency coherence was negatively correlated with autoregulatory gain. TVCF in the frequency range from 0.1 to 0.3 Hz was significantly higher in SDR (7 out of 7, >0.5) than in SHR (5 out of 6, <0.5), and consistent for all time points. For TGF frequency range (0.03-0.05 Hz), coherence exhibited substantial nonstationarity in both strains. Five of six SHR had mean coherence (<0.5), while four of seven SDR exhibited coherence (<0.5). Together, these results demonstrate substantial nonstationarity in autoregulatory dynamics in both SHR and SDR. Furthermore, they indicate that the nonstationarity accounts for most of the dynamic complexity in SDR, but that it accounts for only a part of the dynamic complexity in SHR.

Assessment of Granger causality by nonlinear model identification: application to short-term cardiovascular variability

A method for assessing Granger causal relationships in bivariate time series, based on nonlinear autoregressive (NAR) and nonlinear autoregressive exogenous (NARX) models is presented. The method evaluates bilateral interactions between two time series by quantifying the predictability improvement (PI) of the output time series when the dynamics associated with the input time series are included, i.e., moving from NAR to NARX prediction. The NARX model identification was performed by the optimal parameter search (OPS) algorithm, and its results were compared to the least-squares method to determine the most appropriate method to be used for experimental data. The statistical significance of the PI was assessed using a surrogate data technique. The proposed method was tested with simulation examples involving short realizations of linear stochastic processes and nonlinear deterministic signals in which either unidirectional or bidirectional coupling and varying strengths of interactions were imposed. It was found that the OPS-based NARX model was accurate and sensitive in detecting imposed Granger causality conditions. In addition, the OPS-based NARX model was more accurate than the least squares method. Application to the systolic blood pressure and heart rate variability signals demonstrated the feasibility of the method. In particular, we found a bilateral causal relationship between the two signals as evidenced by the significant reduction in the PI values with the NARX model prediction compared to the NAR model prediction, which was also confirmed by the surrogate data analysis. Furthermore, we found significant reduction in the complexity of the dynamics of the two causal pathways of the two signals as the body position was changed from the supine to upright. The proposed is a general method, thus, it can be applied to a wide variety of physiological signals to better understand causality and coupling that may be different between normal and diseased conditions.

Building dynamical models from data and prior knowledge: the case of the first period-doubling bifurcation

This paper reviews some aspects of nonlinear model building from data with (gray box) and without (black box) prior knowledge. The model class is very important because it determines two aspects of the final model, namely (i) the type of nonlinearity that can be accurately approximated and (ii) the type of prior knowledge that can be taken into account. Such features are usually in conflict when it comes to choosing the model class. The problem of model structure selection is also reviewed. It is argued that such a problem is philosophically different depending on the model class and it is suggested that the choice of model class should be performed based on the type of a priori available. A procedure is proposed to build polynomial models from data on a Poincaré section and prior knowledge about the first period-doubling bifurcation, for which the normal form is also polynomial. The final models approximate dynamical data in a least-squares sense and, by design, present the first period-doubling bifurcation at a specified value of parameters. The procedure is illustrated by means of simulated examples.

A measurement fusion method for nonlinear system identification using a cooperative learning algorithm

Identification of a general nonlinear noisy system viewed as an estimation of a predictor function is studied in this article. A measurement fusion method for the predictor function estimate is proposed. In the proposed scheme, observed data are first fused by using an optimal fusion technique, and then the optimal fused data are incorporated in a nonlinear function estimator based on a robust least squares support vector machine (LS-SVM). A cooperative learning algorithm is proposed to implement the proposed measurement fusion method. Compared with related identification methods, the proposed method can minimize both the approximation error and the noise error. The performance analysis shows that the proposed optimal measurement fusion function estimate has a smaller mean square error than the LS-SVM function estimate. Moreover, the proposed cooperative learning algorithm can converge globally to the optimal measurement fusion function estimate. Finally, the proposed measurement fusion method is applied to ARMA signal and spatial temporal signal modeling. Experimental results show that the proposed measurement fusion method can provide a more accurate model.

Estimation of time-varying coherence function using time-varying transfer functions

We introduce a new method to estimate reliable time-varying coherence functions (TVCF). The technique is based on our previously developed method to estimate time-varying transfer functions (TVTF), known as the time-varying optimal parameter search algorithm (TVOPS) [1]. The TVCF is estimated by the multiplication of the two TVTFs. The two TVTFs are obtained using signal x as the input and signal y as the output to produce the first TVTF, and signal y as the input and signal x as the output to produce the second TVTF. Demonstration of the feasibility and efficacy of the proposed approach is provided with both simulation examples and application to renal blood flow and pressure data. The proposed approach provides higher time-frequency resolution TVCF than afforded by the short time Fourier transform based TVCF.

Closed-loop nonlinear system identification via the vector optimal parameter search algorithm: application to heart rate baroreflex control

The vector optimal parameter search (VOPS) and the constrained optimal parameter search (COPS) are recently developed algorithms for closed-loop linear system identification. We extend both algorithms to be applicable to a closed-loop nonlinear system, which is characterized by a vector nonlinear autoregressive model. Monte Carlo simulations of nonlinear closed-loop systems were performed to compare the performance of the VOPS to the widely utilized vector least squares (VLS), the COPS and the total least squares (TLS) approaches. The relative error and linear transfer functions are computed to determine the accuracy of each method. The comparative results show that both the VOPS and COPS algorithms provide far superior parameter estimates than does the VLS for all simulation examples considered. The TLS provides better estimates than the VOPS, COPS and VLS when there is only observation noise present in the data. However, the performance of the TLS degrades considerably when the data are corrupted by dynamic noise. The clinical applicability of the two extended methods is examined by applying them to a classical physiological closed-loop system, the heart rate baroreflex. It was found that while both control and blockade of parasympathetic system conditions are dominated by linear dynamics, more nonlinearity was observed in the latter. This observation is statistically supported by the calculation of the mutual information of the data and their surrogates.

A weighted-principal component regression method for the identification of physiologic systems

We introduce a system identification method based on weighted-principal component regression (WPCR). This approach aims to identify the dynamics in a linear time-invariant (LTI) model which may represent a resting physiologic system. It tackles the time-domain system identification problem by considering, asymptotically, frequency information inherent in the given data. By including in the model only dominant frequency components of the input signal(s), this method enables construction of candidate models that are specific to the data and facilitates a reduction in parameter estimation error when the signals are colored (as are most physiologic signals). Additionally, this method allows incorporation of preknowledge about the system through a weighting scheme. We present the method in the context of single-input and multi-input single-output systems operating in open-loop and closed-loop. In each scenario, we compare the WPCR method with conventional approaches and approaches that also build data-specific candidate models. Through both simulated and experimental data, we show that the WPCR method enables more accurate identification of the system impulse response function than the other methods when the input signal(s) is colored.

Quantitative multichannel EEG measure predicting the optimal weaning from ventilator in ICU patients with acute respiratory failure

OBJECTIVE

The objective of this study was to develop a novel quantitative multichannel EEG (qEEG) based analysis method, called Global Field Damping Time (GFDT), in order to detect potential EEG changes of patients admitted to the ICU with acute respiratory failure, and correlate them to the patients' recovery outcome predicting the optimal time-point to disconnect the patient from mechanical ventilation.

METHODS

Twenty-nine adult patients with acute respiratory failure out of 98 admitted to the Intensive Care Unit of Saint Paul General Hospital were enrolled, and among them only 15 completed the study. The patients were classified in 3 groups according to their outcome after 3 months follow-up. The patients were intubated with fraction of inspired oxygen (FiO2) of 100%. Neurological Deficit Scores (NDS) were measured 24 h after intubation to assess patients' neurological condition. Twenty-four hours after patient's intubation, FiO2 was decreased to 40% (weaning session), followed by a 5 min early recovery session, a 5 min recovery 1 session and a 5 min recovery 2 session. EEG recordings were performed during this experimental procedure. Multichannel EEG segments were processed and fitted into a multivariate autoregressive (mAR) model, and single channel EEG segments into a scalar autoregressive (sAR) model. The mAR and the sAR models of arbitrary order p were decomposed into mp and p oscillators and relaxators, respectively. Damping time of each oscillator and each relaxator, and the Global Field Damping Time (GFDT) as a weighted damping time were estimated for both mAR and sAR models.

RESULTS

A statistically significant increase of mAR model's GFDT during the weaning session was observed in the subjects of all groups. Comparing the 3 patients' groups, statistically significant differences for mAR model's GFDT were observed for the weaning and early recovery session. Linear regression analysis between NDS and mean mAR model's GFDT showed statistical significance during weaning session, early recovery session, and recovery 1 session. There was no statistical significance for SaO2 in the regression analysis with NDS. The sAR model's GFDT presented worst results in comparison with the mAR modelling GFDT in the identification of hypoxic conditions during weaning session and in the discrimination of patients with acute respiratory failure according to their neurological outcome.

CONCLUSIONS

Global Field Damping Time as correlated to the patients' neurological outcome appears to be a simple, compact, and substantial novel indicator of cerebral hypoxia and a potential predictor of the optimal time-point to disconnect the patient from the ventilator.

SIGNIFICANCE

Quantitative EEG seems to be an important tool for ICU clinicians assisting them to decide for the patients' optimal time-point to disconnect the patient from the ventilator.

Interactions between TGF-dependent and myogenic oscillations in tubular pressure and whole kidney blood flow in both SDR and SHR

We previously showed that nonlinear interactions between the two renal autoregulatory mechanics (tubuloglomerular feedback and the myogenic mechanism) were observed in the stop flow pressure (SFP) and whole kidney blood flow data from Sprague-Dawley rats (SDR) using time-invariant bispectrum analysis ( 3 , 4 ). No such nonlinear interactions were observed in either SFP or whole kidney blood flow data obtained from spontaneously hypertensive rats (SHR). We speculated that the failure to detect nonlinear interactions in the SHR data may be related to our observation that these interactions were not continuous and therefore had time-varying characteristics. Thus the absence of such nonlinear interactions may be due to an inappropriate time-invariant method being applied to data that are especially time varying in nature. We examine this possibility in this paper by using a time-varying bispectrum approach, which we developed for this purpose. Indeed, we found significant nonlinear interactions in SHR ( n = 18 for SFP; n = 12 for whole kidney blood flow). Moreover, the duration of nonlinear coupling is found statistically to be longer ( P = 0.001) in SFP data from either SDR or SHR than it is in whole kidney data from either type of rat. We conclude that nonlinear coupling is present at both the single nephron as well as the whole kidney level for SDR and SHR. In addition, SHR data at the whole kidney level exhibit the most transient nonlinear coupling phenomena.

Estimation of Time-Varying Coherence Function Using Time-Varying Transfer Functions

We introduce a new method to estimate reliable time-varying coherence functions (TVCF) for causal systems. The technique is based on our previously developed method to estimate time-varying transfer functions (TVTF), known as the time-varying optimal parameter search algorithm [Zou, R., H. Wang, and K. H. Chon. A robust time-varying identification algorithm using basis functions. Ann. Biomed. Eng. 31: 840-853, 2003]. The TVCF is estimated by the multiplication of two TVTFs. The two TVTFs are obtained using signal x as the input and signal y as the output to produce the first TVTF, and signal y as the input and signal x as the output to produce the second TVTF. Demonstration of the feasibility and efficacy of the proposed approach is provided with both simulation examples and application to renal blood flow and pressure data. The proposed approach provides higher time-frequency resolution TVCF than afforded by the short time Fourier transform based TVCF.

Nonlinear statistical modeling and model discovery for cardiorespiratory data

We present a Bayesian dynamical inference method for characterizing cardiorespiratory (CR) dynamics in humans by inverse modeling from blood pressure time-series data. The technique is applicable to a broad range of stochastic dynamical models and can be implemented without severe computational demands. A simple nonlinear dynamical model is found that describes a measured blood pressure time series in the primary frequency band of the CR dynamics. The accuracy of the method is investigated using model-generated data with parameters close to the parameters inferred in the experiment. The connection of the inferred model to a well-known beat-to-beat model of the baroreflex is discussed.

Using data-driven discrete-time models and the unscented Kalman filter to estimate unobserved variables of nonlinear systems

This paper addresses the problem of state estimation for nonlinear systems by means of the unscented Kalman filter (UKF). Compared to the traditional extended Kalman filter, the UKF does not require the local linearization of the system equations used in the propagation stage. Important results using the UKF have been reported recently but in every case the system equations used by the filter were considered known. Not only that, such models are usually considered to be differential equations, which requires that numerical integration be performed during the propagation phase of the filter. In this paper the dynamical equations of the system are taken to be difference equations--thus avoiding numerical integration--and are built from data without prior knowledge. The identified models are subsequently implemented in the filter in order to accomplish state estimation. The paper discusses the impact of not knowing the exact equations and using data-driven models in the context of state and joint state-and-parameter estimation. The procedure is illustrated by means of examples that use simulated and measured data.

Identification of Transient Renal Autoregulatory Mechanisms Using Time-Frequency Spectral Techniques

Identification of the two principal mediators of renal autoregulation from time-series data is difficult, as both the tubuloglomerular feedback (TGF) and myogenic (MYO) mechanisms interact and share a common effector, the afferent arteriole. Moreover, although both mechanisms can exhibit oscillations in well-characterized frequency bands, these systems often operate in nonoscillatory states not detectable by frequency-domain analysis. To overcome these difficulties, we have developed a new approach to the characterization of the TGF and MYO systems. A laser Doppler probe is used to measure fluctuations in local cortical blood flow (CBF) in response to spontaneous changes in blood pressure (BP) and to large imposed perturbations in BP, which elicit strong, simultaneous, transient, oscillatory blood flow responses. These transient responses are identified by high-resolution time-frequency spectral analysis of the time-series data. In this report, we compare four different time-frequency spectral techniques (the short-time Fourier transform (STFT), smoothed pseudo Wigner-Ville, and two recently developed methods: the Hilbert-Huang transform and time varying optimal parameter search (TVOPS)) to determine which of these four methods is best suited for the identification of transient oscillations in renal autoregulatory mechanisms. We found that TVOPS consistently provided the best performance in both simulation examples and identification of the two autoregulatory mechanisms in actual data. While the STFT suffers in time and frequency resolution as compared to the other three methods, it was able to identify the two autoregulatory mechanisms. Taken together, our experience suggests a two level approach to the analysis of renal blood flow (RBF) data: STFT to obtain a low-resolution time-frequency spectrogram, followed by the use of a higher resolution technique, such as the TVOPS, if even higher time-frequency resolution of the transient responses is required.

Multiple Time-Varying Dynamic Analysis Using Multiple Sets of Basis Functions

We extend a recently developed algorithm that expands the time-varying parameters onto a single set of basis functions, to multiple sets of basis functions. This feature allows the capability to capture many different dynamics that may be inherent in the system. A single set of basis functions that has its own unique characteristics can best capture dynamics of the system that have similar features. Therefore, for systems that have multiple dynamics, the use of a single set of basis functions may not be adequate. Computer simulation examples do indeed show the benefit of using multiple sets of basis functions over the single set of basis functions for cases with many switching dynamics. Moreover, the proposed method remains accurate even under significant noise contamination. Application of the proposed approach to blood pressure data likewise indicate better tracking capability of the two sets of basis function than the recursive least squares or a single set of basis functions.

Inference of a nonlinear stochastic model of the cardiorespiratory interaction

We reconstruct a nonlinear stochastic model of the cardiorespiratory interaction in terms of a set of polynomial basis functions representing the nonlinear force governing system oscillations. The strength and direction of coupling and noise intensity are simultaneously inferred from a univariate blood pressure signal. Our new inference technique does not require extensive global optimization, and it is applicable to a wide range of complex dynamical systems subject to noise.

A Model Order Selection Criterion With Applications to Cardio-Respiratory-Renal Systems

We introduce a model order selection criterion called signal prediction error (SPE) for the identification of a linear regression model, which can be an adequate representation of a resting physiologic system. SPE is an estimate of the prediction error variance due only to model estimation error and not unobserved noise, which distinguishes it from the widely used final prediction error (FPE). We then present a theoretical analysis of SPE, which predicts that its ability to select correctly the model order is more dependent on the signal-to-noise ratio (SNR) and less dependent on the number of data samples available for analysis. We next propose a heuristic procedure based on SPE (called SPE(D)) to improve its robustness to SNR levels. We then demonstrate, through simulated physiologic data at high SNR levels, that SPE will be equivalent to consistent model order selection criteria for long data records but will become superior to FPE and other model order selection criteria as the size of the data record decreases. The simulated data results also show that SPE(D) is indeed a significant improvement over SPE in terms of robustness to SNR. Finally, we demonstrate the applicability of SPE and SPE(D) to actual cardio-respiratory-renal data.

Interactions of TGF-dependent and myogenic oscillations in tubular pressure

We have previously shown that there are two oscillating components in spontaneously fluctuating single-nephron blood flow obtained from Sprague-Dawley rats (Yip K-P, Holstein-Rathlou NH, and Marsh DJ. Am J Physiol Renal Physiol 264: F427–F434, 1993). The slow oscillation (20–30 mHz) is mediated by tubuloglomerular feedback (TGF), whereas the fast oscillation (100 mHz) is probably related to spontaneous myogenic activity. The fast oscillation is rarely detected in spontaneous tubular pressure because of its small magnitude and the fact that tubular compliance filters pressure waves. We detected myogenic oscillation superimposed on TGF-mediated oscillation when ambient tubular flow was interrupted. Two well-defined peaks are present in the mean power spectrum of stop-flow pressure (SFP) centering at 25 and 100 mHz ( n = 13), in addition to a small peak at 125–130 mHz. Bispectral analysis indicates that two of these oscillations (30 and 100 mHz) interact nonlinearly to produce the third oscillation at 125–130 mHz. The presence of nonlinear interactions between TGF and myogenic oscillations indicates that estimates of the relative contribution of each of these mechanisms in renal autoregulation need to account for this interaction. The magnitude of myogenic oscillations was considerably smaller in the SFP measured from spontaneously hypertensive rats (SHR, n = 13); consequently, nonlinear interactions were not observed with bispectral analysis. Reduced augmentation of myogenic oscillations in SFP of SHR might account for the failure in detecting nonlinear interactions in SHR.

Stability analysis of sleep apnea time series using identified models: a case study

This paper investigates the use of identified nonlinear multivariable autonomous models in the classification of breathing patterns of a patient with sleep apnea. Details about the identification procedure are provided and the results reported for the case study at hand suggest that identified models could be useful in computer-based monitoring.

Robust Algorithm for Estimation of Time-Varying Transfer Functions

We introduce a new method to estimate reliable time-varying (TV) transfer functions (TFs) and TV impulse response functions. The method is based on TV autoregressive moving average models in which the TV parameters are accurately obtained using the optimal parameter search method which we have previously developed. The new method is more accurate than the recursive least-squares (RLS), and remains robust even in the case of significant noise contamination. Furthermore, the new method is able to track dynamics that change abruptly, which is certainly a deficiency of the RLS. Application of the new method to renal blood pressure and flow revealed that hypertensive rats undergo more complex and TV autoregulation in maintaining stable blood flow than do normotensive rats. This observation has not been previously revealed using time-invariant TF analyses. The newly developed approach may promote the broader use of TV system identification in studies of physiological systems and makes linear and nonlinear TV modeling possible in certain cases previously thought intractable.