Surrogate Data Sets Research Articles

A Bayesian Algorithm for Reconstructing Climate Anomalies in Space and Time. Part II: Comparison with the Regularized Expectation–Maximization Algorithm

AbstractPart I presented a Bayesian algorithm for reconstructing climate anomalies in space and time (BARCAST). This method involves specifying simple parametric forms for the spatial covariance and temporal evolution of the climate field as well as “observation equations” describing the relationships between the data types and the corresponding true values of the climate field. As this Bayesian approach to reconstructing climate fields is new and different, it is worthwhile to compare it in detail to the more established regularized expectation–maximization (RegEM) algorithm, which is based on an empirical estimate of the joint data covariance matrix and a multivariate regression of the instrumental time series onto the proxy time series. The differing assumptions made by BARCAST and RegEM are detailed, and the impacts of these differences on the analysis are discussed. Key distinctions between BARCAST and RegEM include their treatment of spatial and temporal covariance, the prior information that enters into each analysis, the quantities they seek to impute, the end product of each analysis, the temporal variance of the reconstructed field, and the treatment of uncertainty in both the imputed values and functions of these imputations. Differences between BARCAST and RegEM are illustrated by applying the two approaches to various surrogate datasets. If the assumptions inherent to BARCAST are not strongly violated, then in scenarios comparable to practical applications BARCAST results in reconstructions of both the field and the spatial mean that are more skillful than those produced by RegEM, as measured by the coefficient of efficiency. In addition, the uncertainty intervals produced by BARCAST are narrower than those estimated using RegEM and contain the true values with higher probability.

A Bayesian Algorithm for Reconstructing Climate Anomalies in Space and Time. Part I: Development and Applications to Paleoclimate Reconstruction Problems

AbstractReconstructing the spatial pattern of a climate field through time from a dataset of overlapping instrumental and climate proxy time series is a nontrivial statistical problem. The need to transform the proxy observations into estimates of the climate field, and the fact that the observed time series are not uniformly distributed in space, further complicate the analysis. Current leading approaches to this problem are based on estimating the full covariance matrix between the proxy time series and instrumental time series over a “calibration” interval and then using this covariance matrix in the context of a linear regression to predict the missing instrumental values from the proxy observations for years prior to instrumental coverage.A fundamentally different approach to this problem is formulated by specifying parametric forms for the spatial covariance and temporal evolution of the climate field, as well as “observation equations” describing the relationship between the data types and the corresponding true values of the climate field. A hierarchical Bayesian model is used to assimilate both proxy and instrumental datasets and to estimate the probability distribution of all model parameters and the climate field through time on a regular spatial grid. The output from this approach includes an estimate of the full covariance structure of the climate field and model parameters as well as diagnostics that estimate the utility of the different proxy time series.This methodology is demonstrated using an instrumental surface temperature dataset after corrupting a number of the time series to mimic proxy observations. The results are compared to those achieved using the regularized expectation–maximization algorithm, and in these experiments the Bayesian algorithm produces reconstructions with greater skill. The assumptions underlying these two methodologies and the results of applying each to simple surrogate datasets are explored in greater detail in Part II.

Modeling Link Events in High Reliability Networks With Support Vector Machines

High reliability communication networks (HRCN) are characterized by very low failure rates, and extremely short unavailability periods. The accurate modeling of the link availability in HRCN is a non-trivial problem, given that an extremely low number of historic events have been observed. We propose a statistical learning model for link event prediction in HRCN based on support vector machines (SVM) for nonlinear regression. The model flexibility can be improved by grouping predictor variables of different nature. A surrogate data set is made, which mimics the basic properties of links in a real network, and it is used for simulations that yield basic knowledge about the use and performance of the proposed SVM model. A true network example, based on two years of historic data, is also analysed. The proposed SVM model yields better performance than other tested methods (frequentist, and neural network estimators), specially during the first years obtaining the historic data in a HRCN, when the number of events is critically low.

Neuroanalysis.Org: Information-Theoretic and Extended Analyses of Neural Coding

Analyses of neural coding-the representation and processing of spike train information-require multiple methods, because neural systems use many kinds of representations, and many analytic methods require specific types or amounts of data. Complementing widely-available conventional methods, we developed and released open source the following via neuroanalysis.org (Goldberg et al. Neuroinformatics 7, 165-178, 2009):• the downloadable STAToolkit suite of information-theoretic algorithms,• guidance for algorithmic development, use, and applicability to neural systems, and• an in-development AnalysisServer.Toward aiding formulation and tests of hypotheses about neural coding, sensory discrimination, firing pattern variability, synchrony and other relations characterizing multineuronal recordings, we now report expanding capabilities of neuroanalysis.org. Utilizing algorithms, code, and/or insight from twelve collaborators, we expand our set of complementary analytic methods, including additional information measures, dimensional reduction, distinguishing information from purely biophysical variation, and generation of surrogate multineuronal data sets. These include a new entropy estimator (our eighth) the ‘NSB’ method, aiding analyses of highly undersampled data. Examples and demonstrations serve to inform and guide neurophysiologists to select STAToolkit methods.Recent publications and presentations describe STAToolkit methodologies, report a new index characterizing how population activity deviates from maximum-entropy models, and discuss applicability of metrics to synaptic processing and large numbers of neurons.To assist computationally-intensive explorations of databased spike trains, we also offer an in-development AnalysisServer, an open-access dedicated large-scale computational array. ‘Analyze’ links will enable dataset grouping, parsing, concatenation, and entropy and information determination from data at neurodatabase.org.Parallel enhancements to neurodatabase.org are intended to aid understanding of neural function in both normal and disease states. We continue to accommodate and solicit an expanding set of data types including recordings from multiple preparations.Supported by Human Brain Project/Neuroinformatics MH068012 from NIMH, NINDS, NIA, NIBIB and NSF.

Extraction and comparison of gene expression patterns from 2D RNA in situ hybridization images

Recent advancements in high-throughput imaging have created new large datasets with tens of thousands of gene expression images. Methods for capturing these spatial and/or temporal expression patterns include in situ hybridization or fluorescent reporter constructs or tags, and results are still frequently assessed by subjective qualitative comparisons. In order to deal with available large datasets, fully automated analysis methods must be developed to properly normalize and model spatial expression patterns. We have developed image segmentation and registration methods to identify and extract spatial gene expression patterns from RNA in situ hybridization experiments of Drosophila embryos. These methods allow us to normalize and extract expression information for 78,621 images from 3724 genes across six time stages. The similarity between gene expression patterns is computed using four scoring metrics: mean squared error, Haar wavelet distance, mutual information and spatial mutual information (SMI). We additionally propose a strategy to calculate the significance of the similarity between two expression images, by generating surrogate datasets with similar spatial expression patterns using a Monte Carlo swap sampler. On data from an early development time stage, we show that SMI provides the most biologically relevant metric of comparison, and that our significance testing generalizes metrics to achieve similar performance. We exemplify the application of spatial metrics on the well-known Drosophila segmentation network. A Java webstart application to register and compare patterns, as well as all source code, are available from: http://tools.genome.duke.edu/generegulation/image_analysis/insitu uwe.ohler@duke.edu Supplementary data are available at Bioinformatics online.

Superstatistical fluctuations in time series: Applications to share-price dynamics and turbulence

We report a general technique to study a given experimental time series with superstatistics. Crucial for the applicability of the superstatistics concept is the existence of a parameter beta that fluctuates on a large time scale as compared to the other time scales of the complex system under consideration. The proposed method extracts the main superstatistical parameters out of a given data set and examines the validity of the superstatistical model assumptions. We test the method thoroughly with surrogate data sets. Then the applicability of the superstatistical approach is illustrated using real experimental data. We study two examples, velocity time series measured in turbulent Taylor-Couette flows and time series of log returns of the closing prices of some stock market indices.

Detection of low-dimensional chaos in buildings energy consumption time series

Abstract In this paper, nonlinear time series modeling techniques are applied to analyze building energy consumption data. The time series were obtained for the benchmark data set Proben 1, and comes from the first energy prediction contest, the Great Building Energy Predictor Shootout I, organized by ASHRAE. The phase space, which describes the evolution of the behavior of a nonlinear system, is reconstructed using the delay embedding theorem suggested by TAKENS. The embedding parameters, e.g. the delay time and the embedding dimension are estimated using the average mutual information (AMI) of the data and the false nearest neighbor (FNN) algorithm, respectively. Nonlinearity was detected, by applying the surrogate data sets method. Numerically estimated non-integral fractal dimension and a positive Lyapunov exponent are not necessarily sufficient indication of chaos; therefore we apply a more stringent criterion, developed by Gao and Zheng, which is based on the logarithmic displacement of time-dependent exponent curves, and show that these data are chaotic. Based on this analysis and proof, we then calculate the correlation dimension of the resulting attractor and the largest Lyapunov exponent. The correlation dimension 3.47 and largest Lyapunov exponent 0.047 are estimated. These results indicate that chaotic characteristics obviously exist in the specific energy consumption data set, and thus techniques based on phase space dynamics can be used to analyze and predict buildings energy use.

Functional clustering algorithm for the analysis of dynamic network data.

We formulate a technique for the detection of functional clusters in discrete event data. The advantage of this algorithm is that no prior knowledge of the number of functional groups is needed, as our procedure progressively combines data traces and derives the optimal clustering cutoff in a simple and intuitive manner through the use of surrogate data sets. In order to demonstrate the power of this algorithm to detect changes in network dynamics and connectivity, we apply it to both simulated neural spike train data and real neural data obtained from the mouse hippocampus during exploration and slow-wave sleep. Using the simulated data, we show that our algorithm performs better than existing methods. In the experimental data, we observe state-dependent clustering patterns consistent with known neurophysiological processes involved in memory consolidation.

Model-Independent Test for Scale-Dependent Non-Gaussianities in the Cosmic Microwave Background

We present a model-independent method to test for scale-dependent non-Gaussianities in combination with scaling indices as test statistics. Therefore, surrogate data sets are generated, in which the power spectrum of the original data is preserved, while the higher order correlations are partly randomized by applying a scale-dependent shuffling procedure to the Fourier phases. We apply this method to the Wilkinson Microwave Anisotropy Probe data of the cosmic microwave background and find signatures for non-Gaussianities on large scales. Further tests are required to elucidate the origin of the detected anomalies.

Multislice multiecho T2* cardiac magnetic resonance for the detection of heterogeneous myocardial iron distribution in thalassaemia patients

The present study investigated myocardial T2* heterogeneity in thalassaemia major (TM) patients by cardiac magnetic resonance (CMR), to determine whether is related to inhomogeneous iron overload distribution. A total of 230 TM patients consecutively referred to our laboratory were studied retrospectively. Three short-axis views (basal, medium and apical) of the left ventricle (LV) were obtained by multislice multiecho T2* CMR. T2* segmental distribution was mapped on a 16-segment LV model. The level of heterogeneity of the T2* segmental distribution, evaluated by the coefficient of variation (CoV), was compared with that of a surrogate data set, to determine whether the inhomogeneous segmental distribution of T2* could be generated by susceptibility artefacts. Susceptibility artefacts offer an explanation for the T2* heterogeneity observed in patients without iron overload. In subjects with global T2* below the lower limit of the normal, T2* heterogeneity increased abruptly which could not be explained by artefactual effects. Some segmental T2* values were below and others above the limit of normal threshold (20 ms) in 104 (45%) TM patients. Among these patients, 74% showed a normal T2* global value. In conclusion, a true heterogeneity in the iron overload distribution may be present in TM patients. Heterogeneity seemingly appears in the borderline myocardial iron and stabilizes at moderate to severe iron burden.

Nonlinear EEG analysis in early Alzheimer's disease.

Nonlinear EEG analysis attempts to characterize the dynamics of neural networks in the brain. Abnormalities in nonlinear EEG measures have been found repeatedly in Alzheimer's disease (AD). The present study was undertaken to investigate whether these abnormalities could already be found in the early stage of AD. In a representative sample of 49 community-dwelling elderly, Alzheimer's disease was diagnosed in 7 subjects. Correlation dimension (D2) and nonlinear prediction were measured at 16 electrodes and in two different activational states. Also, 10 surrogate data sets were generated for each EEG epoch in order to investigate the presence of nonlinear dynamics. Differences between nonlinear statistics derived from original and from surrogate data sets were expressed as Z-scores. We found lower D2 and higher predictability in the demented subjects compared to the normal subjects. The results obtained with the Z-scores pointed to changed nonlinear dynamics in frontal and temporal areas in demented subjects. However, the major differences between demented and healthy subjects are not due to nonlinearity. From this it appears that linear dynamics change first in the course of AD, followed by changes in nonlinear dynamics.

Statistical Significance of Precisely Repeated Intracellular Synaptic Patterns

Can neuronal networks produce patterns of activity with millisecond accuracy? It may seem unlikely, considering the probabilistic nature of synaptic transmission. However, some theories of brain function predict that such precision is feasible and can emerge from the non-linearity of the action potential generation in circuits of connected neurons. Several studies have presented evidence for and against this hypothesis. Our earlier work supported the precision hypothesis, based on results demonstrating that precise patterns of synaptic inputs could be found in intracellular recordings from neurons in brain slices and in vivo. To test this hypothesis, we devised a method for finding precise repeats of activity and compared repeats found in the data to those found in surrogate datasets made by shuffling the original data. Because more repeats were found in the original data than in the surrogate data sets, we argued that repeats were not due to chance occurrence. Mokeichev et al. (2007) challenged these conclusions, arguing that the generation of surrogate data was insufficiently rigorous. We have now reanalyzed our previous data with the methods introduced from Mokeichev et al. (2007). Our reanalysis reveals that repeats are statistically significant, thus supporting our earlier conclusions, while also supporting many conclusions that Mokeichev et al. (2007) drew from their recent in vivo recordings. Moreover, we also show that the conditions under which the membrane potential is recorded contributes significantly to the ability to detect repeats and may explain conflicting results. In conclusion, our reevaluation resolves the methodological contradictions between Ikegaya et al. (2004) and Mokeichev et al. (2007), but demonstrates the validity of our previous conclusion that spontaneous network activity is non-randomly organized.

Multislice Multiecho T2* Cardiovascular Magnetic Resonance Detects Heterogeneous Myocardial Iron Distribution in Thalassemia Patients

Segmental distribution of T2* values can be assessed in heart iron overloaded patients by multislice, multiecho cardiovascular MRI. A significant heterogeneity in T2* segmental distribution was demonstrated in previous studies in thalassemia major (TM) patients [1] and may represent true heterogeneous iron density or could be generated by geometric and susceptibility artefacts. In this study we investigate the relationship between T2* heterogeneity and iron overload progression in a large patient population in order to understand if susceptibility artefact may account for inhomogeneous T2* values segmental distribution.230 TM patients consecutively affered to our laboratory were studied. Informed consent was obtained for all of them. MRI was performed using a 1.5 T MR scanner (GE Signa, CV/i). For the measurements of myocardial T2*, a fast gradient echo-multiecho sequence (FA=25°, matrix=256×192, FOV=35×35 cm, thickness=8.0 mm, NEX=0.75) was used with ECG triggering. Each slice was acquired at nine echo times (2.2–20.3 ms, with echo spacing of 2.26 ms) in a single end-expiratory breath-hold. Three short axis views (basal, medium, and apical) of the left ventricle were obtained and analyzed using a custom-written, previous validated software (HIPPO MIOT®) The myocardium was automatically segmented into a 16-segments standardized heart model and the T2* value on each segment was calculated as well as the global T2* value. The level of heterogeneity of the T2* segmental distribution on each patient was evaluated by computing the coefficient of variation (CoV) as the standard deviation of the absolute value of differences between the segmental T2* values and the global T2* value divided by their means, and expressed as the percentage. A surrogate data sets was obtained stating that the inhomogeneous segmental distribution of T2* would be generated only by susceptibility artefacts that are additive in the R2* (1/T2*) domain. These artefacts were characterized in a previous study by the analysis of the segmental T2* distribution in normal subject [2].In 45 (20%) TM patients, segmental T2* values were all below the lower limit of normal (20 ms). In 104 (45%) patients, T2* values were heterogeneous with respect to the normal threshold. Of these patients, 74% showed a normal T2* global value. Eighty-one (35%) patients had all normal segments. Figure 1 shows the distribution of heterogeneous T2* among the patients population. Figure 2 shows the CoV in myocardial T2* distribution vs. the global T2* values in patients and surrogate data. Patients were sorted using the global T2* values and averaged on a 20-point window to smooth the resulting graph. Mean and normal global T2* value for healthy subjects were assessed in a previous study. Montecarlo simulation was performed on 10,000 surrogate data sets. For surrogate data, the mean CoV values together with the mean⊠2SD limits are shown. T2* heterogeneity for patients without iron overload was compatible with the hypothesis that the heterogeneity was generated by additive susceptibility artefacts. Below the normal limit of global T2* the heterogeneity abruptly increased of about 10%. Starting from this level, the heterogeneity decreased linearly returning to 25% in patients with high iron overload levels. Figure 2 showsIn conclusion, a large percentage of thalassemia major patients shows a significant heterogeneity of segmental distribution of T2* values in myocardium, measured by multislice multiecho T2* cardiac magnetic resonance. T2* segmental heterogeneity in TM patients cannot be explained by the effect of susceptibility artefacts, that are additive and should vanish at high iron overload levels. A possible interpretation is that a true heterogeneity in iron overload distribution is present in TM patients. This heterogeneity seems more important in the early development of iron overload and reduces when the iron overload level increases. [Display omitted] [Display omitted]

Effect of heart failure on cardiac and respiratory variability and cardioventilatory coupling

Changes in heart rate (HRV) and respiratory variabilities are associated with worsening heart failure (HF). We hypothesized that changes in non‐linear dynamic properties of respiratory pattern track the development of early HF. In Sprague Dawley rats (n=6, 3 with sham surgery) were implanted with wireless blood pressure (BP) transducers (DSI transmitters) and the left main cardiac artery was ligated. Heart rate (HR), BP and respiration was monitored simultaneously by placing a plethysmograph on the DSI‐receiver. Animals were exposed to: 21%, 100% and 8% O2 each wk following surgery. Animals were echoed at 6 wk to assess cardiac function. Breath‐to‐breath variability (Ti and Te) was assessed by systematically applying non‐linear analytical tools to epochs of stationary data, and compared the results to 19 surrogate data sets, in which the auto‐correlation function was preserve but the time dependent properties of the data set were destroyed. Mutual information increased and sample entropy of the interval decreased from 3 to 8 wks post‐surgery in HF animals compared to animals those received sham surgery. These changes in variability were especially apparent when animals were breathing a hypoxic mixture. These preliminary data suggest that nonlinear sources of breathing variability are sensitive to cardiac performance decrease even during the early stages of HF (HL‐080318).

The baroreflex contribution to spontaneous heart rhythm assessed with a mathematical model in rats

In Spontaneously Hypertensive (SHR) and Wistar–Kyoto (WKY) normotensive rats, we quantified the extent to which spontaneous fluctuations of heart period (HP) may be determined from arterial pressure based on linear baroreflex properties. We analyzed time series (30-s length) of low-frequency (< 0.8 Hz) fluctuations of HP and mean arterial pressure obtained during quiet wakefulness, rapid-eye-movement sleep (REMS) and non-rapid-eye-movement sleep (NREMS) as well as a control set of surrogate isospectral data with random phase. HP was modeled as the summed output of two parallel linear transfer functions with arterial pressure as input. The mean square difference between modeled and recorded HP was minimized by varying model parameters. The percentage of time series, in which such difference was lower than half the measured HP variance was significantly lower in REMS (6 ± 1%, SHR; 5 ± 1%, WKY) than either in quiet wakefulness (25 ± 2%, SHR; 35 ± 3%, WKY) or NREMS (33 ± 3%, SHR; 27 ± 3%, WKY), and in quiet wakefulness, it was significantly lower in SHR than in WKY. In surrogate data, these percentages were significantly lower than in recorded data during quiet wakefulness and NREMS, but not during REMS. The extent to which linear baroreflex properties explain spontaneous heart rhythm thus depends on the interaction between the behavioral state and the hypertensive disease, and in REMS, may be accounted for by chance couplings between HP and arterial pressure.

Evidence for time-dependent structures in financial data series over long timescales: Opportunities for dynamic market risk allocation

Market participants have a natural timescale, and they only seek to profit from money-making ideas that have a chance of maturing on a similar timescale to that over which they are measured. Due to the increasingly short-term nature of fund management mandates, opportunities have arisen for those participants willing to take stances over the longer term. In this paper, we outline the rigorous tests that demonstrate the existence of money-making opportunities at long (multi-month to multi-year) timescales, as well as the conceptual arguments, and examples of real opportunities taken. Using the concept of surrogate data sets, we can convincingly reject the concept of independently identically distributed returns on a multi-month to multi-year timescale for stocks and indices, including the Barclays Capital Equity Gilt Study data for the 20th century.

Algorithms for generating surrogate data for sparsely quantized time series

The method of surrogate data is frequently used for a statistical examination of nonlinear properties underlying original data. If surrogate data sets are generated by a null hypothesis that the data are derived by a linear process, a rejection of the hypothesis means that the original data have more complex properties. However, we found that if an algorithm for generating surrogate data, for example, amplitude adjusted Fourier transformed, is applied to sparsely quantized data, there are large discrepancies between their power spectrum and that of the original data in lower frequency regions. We performed some simulations to confirm that these errors often lead to false rejections. In this paper, in order to prevent such drawbacks, we advance an extended hypothesis, and propose two improved algorithms for generating surrogate data that reduce the discrepancies of the power spectra. We also confirm the validity of the two improved algorithms with numerical simulations by showing that the extended null hypothesis can be rejected if the time series is produced from chaotic dynamical systems. Finally, we applied these algorithms for analyzing financial tick data as a real example; then we showed that the extended null hypothesis cannot be rejected because the nonlinear statistics or nonlinear prediction errors exhibited are the same as those of the original financial tick time series.

Nonlinear time series analysis of food intake in the dab and the rainbow trout

Evidence suggests that dab and rainbow trout are able to quickly adjust their food intake to an appropriate level when offered novel diets. In addition day-to-day and meal-to-meal food intake varies greatly and meal timing is plastic. Why this is the case is not clear: Food intake in fish is influenced by many factors, however the hierarchy and mechanisms by which these interact is not yet fully understood. A model of food intake may be helpful to understand these phenomena; to determine model type it is necessary to understand the qualitative nature of food intake. Food intake can be regarded as an autoregressive (AR) time series, as the amount of food eaten at time t will be influenced by previous meals, and this allows food intake to be considered using time series analyses. Here, time series data were analysed using nonlinear techniques to obtain qualitative information from which evidence for the hierarchy of mechanisms controlling food intake may be drawn. Time series were obtained for a group of dab and individuals and a group of rainbow trout for analysis. Surrogate data sets were generated to test several null hypotheses describing linear processes and all proved significantly different to the real data, suggesting nonlinear dynamics. Examination of topography and recurrence diagrams suggested that all series were deterministic and non-stationary. The point correlation dimension (PD2i) suggested low-dimensional dynamics. Our findings suggest therefore that any model of appetite should create output that is deterministic, non-stationary, low-dimensional and having nonlinear dynamics.

Using Ambient Vibrations to Detect Loosening of a Composite-to-Metal Bolted Joint in the Presence of Strong Temperature Fluctuations

We present an approach for detecting damage-induced nonlinearities in structures. The method first involves the creation of surrogate data sets conforming to an appropriate null hypothesis (no damage). The second step is to then compare some nonlinear “feature” extracted from the original data to those extracted from the surrogates. Statistically significant differences suggest evidence in favor of the alternative hypothesis, damage. Using this approach we show how loose connections can be detected using ambient “wave” forcing, conforming to the Pierson-Moskowitz distribution, as the source of excitation. We also demonstrate the ability of this technique to operate without a recorded baseline data set and in the presence of widely varying temperatures. The structure in this case is a thick, composite beam bolted to a steel frame. Data are collected using an optical strain sensing system. For this experiment we are able to reliably detect the presence of a loosened bolt.

Nonlinear Analysis of Blood Cell Flux Fluctuations in the Rat Brain Cortex during Stepwise Hypotension Challenge

Blood flow through a region of interest in the brain cortex (cerebral blood flow (CBF)) as measured by laser-Doppler flowmetry (LDF) shows a complex temporal pattern, which can be either merely random or a manifestation of segmental chaotic dynamics of vasomotion-induced flowmotion in the arterial tree, a deterministic phenomenon; or that of a fractal, self-similar correlation order that emerges from the set of segmental perfusion events on statistical ground. Fractal content (F%) was determined by coarse-graining spectral analysis and their self-similar exponent, H, estimated by bridge-detrended Scaled Windowed Variance (bdSWV) method and a variant of the power Spectral Density method ((low)PSD(w,e)). Chaotic dynamics were assessed by computing the correlation dimension (D(corr)) and the largest Lyapunov exponent (lambda(max)) on unfiltered raw and surrogate datasets. In 10 Sprague-Dawley rats anesthetized by halothane, CBF was measured through the thinned calvarium by the LDF method. Blood pressure (BP) was reduced from 100 to 40 mm Hg in steps of 20 mm Hg maintained for 2 mins by the lower body negative pressure method. Fractal and chaotic patterns coexisted in tissue perfusion. CBF did not show autoregulation. At every BP step, F% remained high (72% to 88%) and was independent of BP. Laser-Doppler flowmetry signals proved to be nonstationary fractional Brownian motions. Their H's by bdSWV and (low)PSD(w,e) (0.29+/-0.006 and 0.25+/-0.012, respectively) were independent of BP. Neither D(corr) nor lambda(max) varied with hypotension. Their values were characteristic of a chaotic system, but surrogate data analysis rendered some of them inconclusive. Hence, CBF fluctuations can be regarded as a robust phenomenon that is not abolished even by sustained hypotension at 40 mm Hg.