combined spatial and non spatial prior for inference on mri time series when modelling fmri and other mri time series data a bayesian approach based on adaptive spatial smoothness priors is a compelling alternative to using a standard generalized linear model glm on presmoothed data another benefit of the bayesian approach is that biophysical prior information can be incorporated in a principled manner however this requirement for a fixed non spatial prior on a parameter would normally preclude using spatial regularization on that same parameter we have developed a gaussian process based prior to apply adaptive spatial regularization while still ensuring that the fixed biophysical prior is correctly applied on each voxel a parameterized covariance matrix provides separate control over the variance the diagonal elements and the between voxel correlation due to off diagonal elements analysis proceeds using evidence optimization eo with variational bayes vb updates used for some parameters the method can also be applied to non linear forward models by using a linear taylor expansion centred on the latest parameter estimates applying the method to fmri with a constrained haemodynamic response function hrf shape model shows improved fits in simulations compared to using either the non spatial or spatial smoothness prior alone we also analyse multi inversion arterial spin labelling data using a non linear perfusion model to estimate cerebral blood flow and bolus arrival time by combining both types of prior information this new prior performs consistently well across a wider range of situations than either prior alone and provides better estimates when both types of prior information are relevant

testing for nonlinearity in unevenly sampled time series we generalize the method of surrogate data of testing for nonlinearity in time series to the case that the data are sampled with uneven time intervals the null hypothesis will be that the data have been generated by a linear stochastic process possibly rescaled and sampled at times chosen independently from the generating process the surrogate data are generated with their linear properties specified by the lomb periodogram the inversion problem is solved by combinatorial optimization

a robust method for detecting interdependences application to intracranially recorded eeg we present a measure for characterizing statistical relationships between two time sequences in contrast to commonly used measures like cross correlations coherence and mutual information the proposed measure is non symmetric and provides information about the direction of interdependence it is closely related to recent attempts to detect generalized synchronization however we do not assume a strict functional relationship between the two time sequences and try to define the measure so as to be robust against noise and to detect also weak interdependences we apply our measure to intracranially recorded electroencephalograms of patients suffering from severe epilepsies

detecting dynamical interdependence and generalized synchrony through mutual prediction in a neural ensemble a method to characterize dynamical interdependence among nonlinear systems is derived based on mutual nonlinear prediction systems with nonlinear correlation will show mutual nonlinear prediction when standard analysis with linear cross correlation might fail mutual nonlinear prediction also provides information on the directionality of the coupling between systems furthermore the existence of bidirectional mutual nonlinear prediction in unidirectionally coupled systems implies generalized synchrony numerical examples studied include three classes of unidirectionally coupled systems systems with identical parameters nonidentical parameters and stochastic driving of a nonlinear system this technique is then applied to the activity of motoneurons within a spinal cord motoneuron pool the interrelationships examined include single neuron unit firing the total number of neurons discharging at one time as measured by the integrated monosynaptic reflex and intracellular measurements of integrated excitatory postsynaptic potentials epsp s dynamical interdependence perhaps generalized synchrony was identified in this neuronal network between simultaneous single unit firings between units and the population and between units and intracellular epsp s

classical and bayesian inference in neuroimaging theory this paper reviews hierarchical observation models used in functional neuroimaging in a bayesian light it emphasizes the common ground shared by classical and bayesian methods to show that conventional analyses of neuroimaging data can be usefully extended within an empirical bayesian framework in particular we formulate the procedures used in conventional data analysis in terms of hierarchical linear models and establish a connection between classical inference and parametric empirical bayes peb through covariance component estimation this estimation is based on an expectation maximization or em algorithm the key point is that hierarchical models not only provide for appropriate inference at the highest level but that one can revisit lower levels suitably equipped to make bayesian inferences bayesian inferences eschew many of the difficulties encountered with classical inference and characterize brain responses in a way that is more directly predicated on what one is interested in the motivation for bayesian approaches is reviewed and the theoretical background is presented in a way that relates to conventional methods in particular restricted maximum likelihood reml this paper is a technical and theoretical prelude to subsequent papers that deal with applications of the theory to a range of important issues in neuroimaging these issues include i estimating nonsphericity or variance components in fmri time series that can arise from serial correlations within subject or are induced by multisubject i e hierarchical studies ii spatiotemporal bayesian models for imaging data in which voxels specific effects are constrained by responses in other voxels iii bayesian estimation of nonlinear models of hemodynamic responses and iv principled ways of mixing structural and functional priors in eeg source reconstruction although diverse all these estimation problems are accommodated by the peb framework described in this paper

synchronization likelihood an unbiased measure of generalized synchronization in multivariate data sets the study of complex systems consisting of many interacting subsystems requires the use of analytical tools which can detect statistical dependencies between time series recorded from these subsystems typical examples are the electroencephalogram eeg and magnetoencephalogram meg which may involve the simultaneous recording of 150 or more time series coherency which is often used to study such data is only sensitive to linear and symmetric interdependencies and cannot deal with non stationarity recently several algorithms based upon the concept of generalized synchronization have been introduced to overcome some of the limitations of coherency estimates e g physica d 134 1999 419 brain res 792 1998 24 however these methods are biased by the degrees of freedom of the interacting subsystems physica d 134 1999 419 physica d 148 2001 147 we propose a novel measure for generalized synchronization in multivariate data sets which avoids this bias and can deal with non stationary dynamics

significance testing of information theoretic functionals abstract a transformation is derived such that under the transformation the mutual information of two unrelated variables is a normally distributed random variable with zero mean and unit variance it is demonstrated that the transformed mutual information can be used as a test statistic to test the statistical significance of the relationship between variables and provides an alternative to the method of surrogate data if the mutual information is regarded as a generalization of the linear correlation coefficient then this test statistic can be regarded as analogous to the fisher z test statistic

population dynamics variance and the sigmoid activation function this paper demonstrates how the sigmoid activation function of neural mass models can be understood in terms of the variance or dispersion of neuronal states we use this relationship to estimate the probability density on hidden neuronal states using non invasive electrophysiological eeg measures and dynamic casual modelling the importance of implicit variance in neuronal states for neural mass models of cortical dynamics is illustrated using both synthetic data and real eeg measurements of sensory evoked responses

eeg fmri fusion of paradigm free activity using kalman filtering we address here the use of eeg and fmri and their combination in order to estimate the full spatiotemporal patterns of activity on the cortical surface in the absence of any particular assumptions on this activity such as stimulation times for handling such a high dimension inverse problem we propose the use of 1 a global forward model of how these measures are functions of the neural activity of a large number of sources distributed on the cortical surface formalized as a dynamical system and 2 adaptive filters as a natural solution to solve this inverse problem iteratively along the temporal dimension this estimation framework relies on realistic physiological models uses eeg and fmri in a symmetric manner and takes into account both their temporal and spatial information we use the kalman filter and smoother to perform such an estimation on realistic artificial data and demonstrate that the algorithm can handle the high dimensionality of these data and that it succeeds in solving this inverse problem combining efficiently the information provided by the two modalities this information being naturally predominantly temporal for eeg and spatial for fmri it performs particularly well in reconstructing a random temporally and spatially smooth activity spread over the cortex the kalman filter and smoother show some limitations however which call for the development of more specific adaptive filters first they do not cope well with the strong nonlinearity in the model that is necessary for an adequate description of the relation between cortical electric activities and the metabolic demand responsible for fmri signals second they fail to estimate a sparse activity i e presenting sharp peaks at specific locations and times finally their computational cost remains high we use schematic examples to explain these limitations and propose further developments of our method to overcome them

direct and indirect optimal control applied to plant virus propagation with seasonality and delays abstract in many applications of mathematical modeling to biology economics social sciences and engineering the objective is to find optimal solutions usually we want to minimize an objective function depending on a number of functions subject to constraints given for example by systems of differential equations two main numerical approaches are used to solve these optimal control problems depending on whether the problem is optimized first and then discretized or vice versa each of these two approaches has its advantages and disadvantages in this paper we describe both methods an apply them to a plant virus propagation model where the virus is propagated through a vector that bites the infected plants the model includes delays due to the time the virus takes to infect the plant and the vector and seasonality due to the dependence of the behavior on the seasons the objective function is the total cost to a farmer of having infected plants and includes the actual cost of a plant plus the cost of the controls which are insecticides and a predator species that preys on the insects numerical simulations are done using both methods and comparisons are made

test your surrogate data before you test for nonlinearity the schemes for the generation of surrogate data in order to test the null hypothesis of linear stochastic process undergoing nonlinear static transform are investigated as to their consistency in representing the null hypothesis in particular we pinpoint some important caveats of the prominent algorithm of amplitude adjusted fourier transform surrogates aaft and compare it to the iterated aaft which is more consistent in representing the null hypothesis it turns out that in many applications with real data the inferences of nonlinearity after marginal rejection of the null hypothesis were premature and have to be reinvestigated taking into account the inaccuracies in the aaft algorithm mainly concerning the mismatching of the linear correlations in order to deal with such inaccuracies we propose the use of linear together with nonlinear polynomials as discriminating statistics the application of this setup to some well known real data sets cautions against the use of the aaft algorithm

bayesian model averaging in eeg meg imaging in this paper the bayesian theory is used to formulate the inverse problem ip of the eeg meg this formulation offers a comparison framework for the wide range of inverse methods available and allows us to address the problem of model uncertainty that arises when dealing with different solutions for a single data in this case each model is defined by the set of assumptions of the inverse method used as well as by the functional dependence between the data and the primary current density pcd inside the brain the key point is that the bayesian theory not only provides for posterior estimates of the parameters of interest the pcd for a given model but also gives the possibility of finding posterior expected utilities unconditional on the models assumed in the present work this is achieved by considering a third level of inference that has been systematically omitted by previous bayesian formulations of the ip this level is known as bayesian model averaging bma the new approach is illustrated in the case of considering different anatomical constraints for solving the ip of the eeg in the frequency domain this methodology allows us to address two of the main problems that affect linear inverse solutions lis a the existence of ghost sources and b the tendency to underestimate deep activity both simulated and real experimental data are used to demonstrate the capabilities of the bma approach and some of the results are compared with the solutions obtained using the popular low resolution electromagnetic tomography loreta and its anatomically constraint version cloreta

a state space model of the hemodynamic approach nonlinear filtering of bold signals in this paper a new procedure is presented which allows the estimation of the states and parameters of the hemodynamic approach from blood oxygenation level dependent bold responses the proposed method constitutes an alternative to the recently proposed friston neuroimage 16 2002 513 method and has some advantages over it the procedure is based on recent groundbreaking time series analysis techniques that have been in this case adopted to characterize hemodynamic responses in functional magnetic resonance imaging fmri this work represents a fundamental improvement over existing approaches to system identification using nonlinear hemodynamic models and is important for three reasons first our model includes physiological noise previous models have been based upon ordinary differential equations that only allow for noise or error to enter at the level of observation secondly by using the innovation method and the local linearization filter not only the parameters but also the underlying states of the system generating responses can be estimated these states can include things like a flow inducing signal triggered by neuronal activation de oxyhemoglobine cerebral blood flow and volume finally radial basis functions have been introduced as a parametric model to represent arbitrary temporal input sequences in the hemodynamic approach which could be essential to understanding those brain areas indirectly related to the stimulus hence thirdly by inferring about the radial basis parameters we are able to perform a blind deconvolution which permits both the reconstruction of the dynamics of the most likely hemodynamic states and also to implicitly reconstruct the underlying synaptic dynamics induced experimentally which caused these states variations from this study we conclude that in spite of the utility of the standard discrete convolution approach used in statistical parametric maps spm nonlinear bold phenomena and unspecific input temporal sequences must be included in the fmri analysis

constrained realization monte carlo method for hypothesis testing abstract we compare two theoretically distinct approaches to generate artificial or surrogate data for testing hypotheses about a given data set the first and more straightforward approach is to fit a single best model to the original data and then to generate surrogate data sets that are typical realizations of that model the second approach concentrates not on the model but directly on the original data it attempts to constrain the surrogate data sets so that they exactly agree with the original data for a specified set of sample statistics examples of these two approaches are provided for two simple cases a test for deviations from a gaussian distribution and a test for serial dependence in a time series additionally we consider test for nonlinearity in time series based on a fourier transform ft method and on more conventional autoregressive moving average arma fits to the data the comparative performance of hypothesis testing schemes based on these two approaches is found to depend on whether or not the discriminating statistic is pivotal a statistic is pivotal if its distribution is the same for all processes consistent with the null hypothesis the typical realization method requires that the discriminating statistic satisfy this property the constrained realization approach on the other hand does not share this requirement and can provide an accurate and powerful test without having to sacrifice flexibility in the choice of discriminating statistic

comparing dynamic causal models this article describes the use of bayes factors for comparing dynamic causal models dcms dcms are used to make inferences about effective connectivity from functional magnetic resonance imaging fmri data these inferences however are contingent upon assumptions about model structure that is the connectivity pattern between the regions included in the model given the current lack of detailed knowledge on anatomical connectivity in the human brain there are often considerable degrees of freedom when defining the connectional structure of dcms in addition many plausible scientific hypotheses may exist about which connections are changed by experimental manipulation and a formal procedure for directly comparing these competing hypotheses is highly desirable in this article we show how bayes factors can be used to guide choices about model structure both concerning the intrinsic connectivity pattern and the contextual modulation of individual connections the combined use of bayes factors and dcm thus allows one to evaluate competing scientific theories about the architecture of large scale neural networks and the neuronal interactions that mediate perception and cognition

on the posynomial fractional programming problems abstract the posynomial fractional programming pfp problem arises from the summation minimization of several quotient terms which are composed of posynomial terms appearing in the objective function subject to given posynomial constraints this paper proposes an approximate approach to solving a pfp problem a linear programming relaxation is derived for the problem based on piecewise linearization techniques which first convert a posynomial term into the sum of absolute terms these absolute terms are then linearized by some linearization techniques the proposed approach could reach a solution as close as possible to a global optimum

a wavelet based method for surrogate data generation hypothesis testing based on surrogate data has emerged as a popular way to test the null hypothesis that a signal is a realisation of a linear gaussian stochastic process if these surrogates are constrained to the values and power spectrum of the original data there is no need to formulate a pivotal test statistic in this paper a method is presented for generating constrained surrogates using a wavelet transform introducing a threshold above which wavelet detail coefficients are pinned to their original values such surrogates avoid problems of nonstationarity for pseudo periodic data and appear to be more robust than conventional approaches for situations where period modulation is corrupting a gaussian stochastic process when used for generating ensemble realisations of a process the approach used here avoids some of the difficulties of methods based on simple randomisation of wavelet coefficients

correlation structure of the discrete wavelet coefficients of fractional brownian motion it is shown that the discrete wavelet coefficients of fractional brownian motion at different scales are correlated and that their auto and cross correlation functions decay hyperbolically fast at a rate much faster than that of the autocorrelation of the fractional brownian motion itself the rate of decay of the correlation function in the wavelet domain is primarily determined by the number of vanishing moments of the analyzing wavelet

bayesian estimation of dynamical systems an application to fmri this paper presents a method for estimating the conditional or posterior distribution of the parameters of deterministic dynamical systems the procedure conforms to an em implementation of a gauss newton search for the maximum of the conditional or posterior density the inclusion of priors in the estimation procedure ensures robust and rapid convergence and the resulting conditional densities enable bayesian inference about the model parameters the method is demonstrated using an input state output model of the hemodynamic coupling between experimentally designed causes or factors in fmri studies and the ensuing bold response this example represents a generalization of current fmri analysis models that accommodates nonlinearities and in which the parameters have an explicit physical interpretation second the approach extends classical inference based on the likelihood of the data given a null hypothesis about the parameters to more plausible inferences about the parameters of the model given the data this inference provides for confidence intervals based on the conditional density 2002 elsevier science usa

modeling plant virus propagation with delays plants play a vital role in the everyday life of all organisms on earth sometimes however plants become infected with a virus this can have a devastating effect on the ecosystem that depends on it an insect vector can cause the transmission of the virus from plant to plant in this paper a system of ordinary differential equations was first used to model the interaction between the insects and the plants we found the equilibria of the model and we analyzed the stability using the reproductive number derived by the next generation matrix afterwards we introduced a delay to the model and a system of delay differential equations dde was obtained analysis of the ddes dynamics was performed including equilibria stability and bifurcations then numerical solutions of the ode and the dde were compared and conclusions are given

statistics of biophysical signal characteristics and state specificity of the human eeg abstract the analysis of records of biophysical signals like eeg have become increasingly interesting to physicists since the development of new algorithms for time series analysis this analysis however requires both an adequate methodology assessing the time series characteristics and an appropriate statistical assessment of the measures resulting from these algorithms the latter should separate inherent effects from those appearing only by chance e g in terms of an appropriate test statistics as a step towards such a statistical assessment we present here a data resampling approach data resampling techniques allow for the estimation of the variance of an f value as obtained through variance analysis a test statistics based on the so called f ratio 2 f f 2 is derived and the simulation of outcomes allows for the determination of its quantiles a multivariate formulation allows for the simultaneous inclusion of linear and nonlinear time series measures into the analysis we applied these concepts to the problem of distinguishing eeg of healthy probands from the eeg of patients diagnosed as schizophrenic together with an approach involving bayes formula to weight the variables it reveals a reliable discriminance performance of 81 when the nonlinear measure and the linear measures and power are included

surrogates for finding unstable periodic orbits in noisy data sets recently searches for unstable periodic orbits in biological and medical applications have become of interest the motivations for this research range in order of ascending complexity from efforts to understand the dynamics of simple sensory neurons through speculations regarding neural coding to the hopeful development of new diagnostic and or control techniques for cardiac and epileptic pathologies biological and medical data are however noisy and nonstationary findings of unstable periodic orbits in such data thus require convincing assessments of their statistical significance such tests are accomplished by comparison with surrogate data files designed to test an appropriate null hypothesis in this paper we test surrogates generated by three different algorithms against correlated noise as well as stable periodic orbits one of the surrogates is new and has been specifically designed to preserve the shape of the attractor we discuss the suitability of these surrogates and argue that the simple shuffled one correctly tests the appropriate null hypothesis

dynamic modeling of neuronal responses in fmri using cubature kalman filtering this paper presents a new approach to inverting fitting models of coupled dynamical systems based on state of the art cubature kalman filtering crucially this inversion furnishes posterior estimates of both the hidden states and parameters of a system including any unknown exogenous input because the underlying generative model is formulated in continuous time with a discrete observation process it can be applied to a wide variety of models specified with either ordinary or stochastic differential equations these are an important class of models that are particularly appropriate for biological time series where the underlying system is specified in terms of kinetics or dynamics i e dynamic causal models we provide comparative evaluations with generalized bayesian filtering dynamic expectation maximization and demonstrate marked improvements in accuracy and computational efficiency we compare the schemes using a series of difficult nonlinear toy examples and conclude with a special focus on hemodynamic models of evoked brain responses in fmri our scheme promises to provide a significant advance in characterizing the functional architectures of distributed neuronal systems even in the absence of known exogenous experimental input e g resting state fmri studies and spontaneous fluctuations in electrophysiological studies importantly unlike current bayesian filters e g dem our scheme provides estimates of time varying parameters which we will exploit in future work on the adaptation and enabling of connections in the brain

statistical parametric mapping for event related potentials ii a hierarchical temporal model in this paper we describe a temporal model for event related potentials erp in the context of statistical parametric mapping spm in brief we project channel data onto a two dimensional scalp surface or into three dimensional brain space using some appropriate inverse solution we then treat the spatiotemporal data in a mass univariate fashion this implicitly factorises the model into spatial and temporal components the key contribution of this paper is the use of observation models that afford an explicit distinction between observation error and variation in the expression of erps this distinction is created by employing a two level hierarchical model in which the first level models the erp effects within subject and trial type while the second models differences in erp expression among trial types and subjects by bringing the analysis of erp data into a classical hierarchical i e mixed effects framework many apparently disparate approaches e g conventional p300 analyses and time frequency analyses of stimulus locked oscillations can be reconciled within the same estimation and inference procedure inference proceeds in the normal way using t or f statistics to test for effects that are localised in peristimulus time or in some time frequency window the use of f statistics is an important generalisation of classical approaches because it allows one to test for effects that lie in a multidimensional subspace i e of unknown but constrained form we describe the analysis procedures the underlying theory and compare its performance to established techniques

measuring statistical dependences in a time series we propose two methods to measure all linear and nonlinear statistical dependences in a stationary time series presuming ergodicity the measures can be obtained from efficient numerical algorithms

spatiotemporal nonlinearity in resting state fmri of the human brain in this work the spatiotemporal nonlinearity in resting state fmri data of the human brain was detected by nonlinear dynamics methods nine human subjects during resting state were imaged using single shot gradient echo planar imaging on a 1 5t scanner eigenvalue spectra for the covariance matrix correlation dimensions and spatiotemporal lyapunov exponents were calculated to detect the spatiotemporal nonlinearity in resting state fmri data by simulating adjusting and comparing the eigenvalue spectra of pure correlated noise with the corresponding real fmri data the intrinsic dimensionality was estimated the intrinsic dimensionality was used to extract the first few principal components from the real fmri data using principal component analysis which will preserve the correct phase dynamics while reducing both computational load and noise level of the data then the phase space was reconstructed using the time delay embedding method for their principal components and the correlation dimension was estimated by the grassberger procaccia algorithm of multiple variable series the spatiotemporal lyapunov exponents were calculated by using the method based on coupled map lattices through nonlinearity testing there are significant differences of correlation dimensions and spatiotemporal lyapunov exponents between fmri data and their surrogate data the fractal dimension and the positive spatiotemporal lyapunov exponents characterize the spatiotemporal nonlinear dynamics property of resting state fmri data therefore the results suggest that fluctuations presented in resting state may be an inherent model of basal neural activation of human brain cannot be fully attributed to noise

variational free energy and the laplace approximation this note derives the variational free energy under the laplace approximation with a focus on accounting for additional model complexity induced by increasing the number of model parameters this is relevant when using the free energy as an approximation to the log evidence in bayesian model averaging and selection by setting restricted maximum likelihood reml in the larger context of variational learning and expectation maximisation em we show how the reml objective function can be adjusted to provide an approximation to the log evidence for a particular model this means reml can be used for model selection specifically to select or compare models with different covariance components this is useful in the context of hierarchical models because it enables a principled selection of priors that under simple hyperpriors can be used for automatic model selection and relevance determination ard deriving the reml objective function from basic variational principles discloses the simple relationships among variational bayes em and reml furthermore we show that em is formally identical to a full variational treatment when the precisions are linear in the hyperparameters finally we also consider briefly dynamic models and how these inform the regularisation of free energy ascent schemes like em and reml

a metropolis hastings algorithm for dynamic causal models abstract dynamic causal modelling dcm is a modelling framework used to describe causal interactions in dynamical systems it was developed to infer the causal architecture of networks of neuronal populations in the brain friston k j harrison l penny w 2003 dynamic causal modelling neuroimage aug 19 4 1273 302 in current formulations of dcm the mean structure of the likelihood is a nonlinear and numerical function of the parameters which precludes exact or analytic bayesian inversion to date approximations to the posterior depend on the assumption of normality i e the laplace assumption in particular two arguments have been used to motivate normality of the prior and posterior distributions first gaussian priors on the parameters are specified carefully to ensure that activity in the dynamic system of neuronal populations converges to a steady state i e the dynamic system is dissipative secondly normality of the posterior is an approximation based on general asymptotic results regarding the form of the posterior under infinite data friston k j harrison l penny w 2003 dynamic causal modelling neuroimage aug 19 4 1273 302 here we provide a critique of these assumptions and evaluate them numerically we use a bayesian inversion scheme the metropolis hastings algorithm that eschews both assumptions this affords an independent route to the posterior and an external means to assess the performance of conventional schemes for dcm it also allows us to assess the sensitivity of the posterior to different priors first we retain the conventional priors and compare the ensuing approximate posterior laplace to the exact posterior mcmc our analyses show that the laplace approximation is appropriate for practical purposes in a second independent set of analyses we compare the exact posterior under conventional priors with an exact posterior under newly defined uninformative priors reassuringly we observe that the posterior is for all practical purposes insensitive of the choice of prior

testing for nonlinearity using redundancies quantitative and qualitative aspects abstract a method for testing nonlinearity in time series is described based on information theoretic functionals redundancies linear and nonlinear forms of which allow either qualitative or after incorporating the surrogate data technique quantitative evaluation of dynamical properties of scrutinized data an interplay of quantitative and qualitative testing on both the linear and nonlinear levels is analyzed and robustness of this combined approach against spurious nonlinearity detection is demonstrated evaluation of redundancies and redundancy based statistics as functions of time lag and embedding dimension can further enhance insight into dynamics of a system under study

wavelets and functional magnetic resonance imaging of the human brain the discrete wavelet transform dwt is widely used for multiresolution analysis and decorrelation or whitening of nonstationary time series and spatial processes wavelets are naturally appropriate for analysis of biological data such as functional magnetic resonance images of the human brain which often demonstrate scale invariant or fractal properties we provide a brief formal introduction to key properties of the dwt and review the growing literature on its application to fmri we focus on three applications in particular i wavelet coefficient resampling or wavestrapping of 1 d time series 2 to 3 d spatial maps and 4 d spatiotemporal processes ii wavelet based estimators for signal and noise parameters of time series regression models assuming the errors are fractional gaussian noise fgn and iii wavelet shrinkage in frequentist and bayesian frameworks to support multiresolution hypothesis testing on spatially extended statistic maps we conclude that the wavelet domain is a rich source of new concepts and techniques to enhance the power of statistical analysis of human fmri data

constrained surrogate time series with preservation of the mean and variance structure a method is presented for generating surrogates that are constrained realizations of a time series but which preserve the local mean and variance of the original signal the method is based on the popular iterated amplitude adjusted fourier transform method but makes use of a wavelet transform to constrain behavior in the time domain using this method it is possible to test for local changes in the nonlinear properties of the signal we present an example for a change in hurst exponent in a time series produced by fractional brownian motion

finite correlation dimension for stochastic systems with power law spectra abstract we discuss a counter example to the traditional view that stochastic processes lead to a non convergence of the correlation dimension in computed or measured time series specifically we show that a simple class of colored random noises characterized by a power law power spectrum have a finite and predictable value for the correlation dimension these results have implications on the experimental study of deterministic chaos as they indicate that the soie observation of a finite fractal dimension from the analysis of a time series is not sufficient to infer the presence of a strange attractor in the system dynamics we demonstrate that the types of random noises considered herein may be given an interpretation in terms of their fractal properties the consequent exploitation of the non gaussian behavior of these random noises leads us to the introduction of a new time series analysis method which we call multivariate scaling analysis we apply this approach to characterize several global properties of random noise

modeling plant virus propagation with seasonality abstract plants are essential to life they are a source of food medicine clothing and are important to a healthy environment unfortunately plants can become infected with a disease a viral infection is one way that a plant may become diseased often times plants die from this infection these viruses hurt the agriculture industry as billions of dollars are lost due to crop loss every year an insect vector is typically the cause for the virus propagation the vectors exhibit seasonal behavior as they are active in the warm months but not as much so in the cooler months to defend against the vectors pesticides have been used while the pesticides might be effective in controlling the vectors they can have harmful side effects on the environment an alternative solution is to introduce a predator or just increase the number of a naturally present one to prey on the insects in this paper we use a mathematical model of ordinary differential equation to model the dynamics of this biological process we first present an autonomous system then two nonautonomous systems accounting for the periodic nature of the insects to analyze the models the basic reproductive number is used we demonstrate a couple of approaches for determining this number a time average approach and a linear operator approach afterwards numerical simulations are used to demonstrate the results finally comparisons are made between the models and the approaches

multilevel linear modelling for fmri group analysis using bayesian inference functional magnetic resonance imaging studies often involve the acquisition of data from multiple sessions and or multiple subjects a hierarchical approach can be taken to modelling such data with a general linear model glm at each level of the hierarchy introducing different random effects variance components inferring on these models is nontrivial with frequentist solutions being unavailable a solution is to use a bayesian framework one important ingredient in this is the choice of prior on the variance components and top level regression parameters due to the typically small numbers of sessions or subjects in neuroimaging the choice of prior is critical to alleviate this problem we introduce to neuroimage modelling the approach of reference priors which drives the choice of prior such that it is noninformative in an information theoretic sense we propose two inference techniques at the top level for multilevel hierarchies a fast approach and a slower more accurate approach we also demonstrate that we can infer on the top level of multilevel hierarchies by inferring on the levels of the hierarchy separately and passing summary statistics of a noncentral multivariate t distribution between them

a wavelet method for the characterization of spatiotemporal patterns this paper introduces a wavelet based method for the characterization of spatiotemporal patterns based on the wavelet multiresolution analysis two wavelet indices multiscale accumulative density mad and multiscale accumulative change mac are proposed for the characterization of the dynamics of the spatiotemporal patterns both indices are constructed by using orthogonal wavelet projection operators the mad is a measure of the spatial complexity of a pattern at a given time whereas the mac characterizes the spatial complexity of instantaneous change of the spatiotemporal patterns at a given time the ratio of the mad indices between the lowest and the highest scales reflects the order of coherence in a pattern the time series of both mad and mac provide the dynamical information of morphological pattern evolutions numerical experiments based on the cahn hilliard equation indicate that the proposed method is efficient for quantitatively characterizing the dynamics of the spatiotemporal patterns 2002 published by elsevier science b v

generalized autoregressive conditional heteroskedasticity a natural generalization of the arch autoregressive conditional heteroskedastic process introduced in engle 1982 to allow for past conditional variances in the current conditional variance equation is proposed stationarity conditions and autocorrelation structure for this new class of parametric models are derived maximum likelihood estimation and testing are also considered finally an empirical example relating to the uncertainty of the inflation rate is presented

nonlinear system analysis and identification from random data 1991 nonlinear system analysis and identification from random data technometrics vol 33 no 4 pp 482 483

filtered noise can mimic low dimensional chaotic attractors this contribution presents four results first calculations indicate that when examined by the grassberger procaccia algorithm alone filtered noise can mimic low dimensional chaotic attractors given the ubiquity of signal filtering in experimental investigations this is potentially important second a criterion is derived which provides an estimate of the minimum data accuracy needed to resolve the dimension of an attractor third it is shown that a criterion derived by eckmann and ruelle physica d 56 185 1992 to estimate the minimum number of data points required in a grassberger procaccia calculation can be used to provide a further check on these dimension estimates fourth it is shown that surrogate data techniques recently published by theiler and his colleagues in nonlinear modeling and forecasting edited by m casdagli and s eubanks addison wesley reading ma 1992 can successfully distinguish between linearly correlated noise and nonlinear structure these results and most particularly the first indicate that grassberger procaccia results must be interpreted with far greater circumspection than has previously been the case and that the algorithm should be used in combination with additional procedures such as calculations with surrogate data when filtered signals are examined by this algorithm alone a finite noninteger value of mathit d 2 is consistent with low dimensional chaotic behavior but it is certainly not a definitive diagnostic of chaos

improvements to surrogate data methods for nonstationary time series the method of surrogate data has been extensively applied to hypothesis testing of system linearity when only one realization of the system a time series is known normally surrogate data should preserve the linear stochastic structure and the amplitude distribution of the original series classical surrogate data methods such as random permutation amplitude adjusted fourier transform or iterative amplitude adjusted fourier transform are successful at preserving one or both of these features in stationary cases however they always produce stationary surrogates hence existing nonstationarity could be interpreted as dynamic nonlinearity certain modifications have been proposed that additionally preserve some nonstationarity at the expense of reproducing a great deal of nonlinearity however even those methods generally fail to preserve the trend i e global nonstationarity in the mean of the original series this is the case of time series with unit roots in their autoregressive structure additionally those methods based on fourier transform either need first and last values in the original series to match or they need to select a piece of the original series with matching ends these conditions are often inapplicable and the resulting surrogates are adversely affected by the well known artefact problem in this study we propose a simple technique that applied within existing fourier transform based methods generates surrogate data that jointly preserve the aforementioned characteristics of the original series including even strong trends moreover our technique avoids the negative effects of end mismatch several artificial and real stationary and nonstationary linear and nonlinear time series are examined in order to demonstrate the advantages of the methods corresponding surrogate data are produced with the classical and with the proposed methods and the results are compared

a branch and cut algorithm for a class of sum of ratios problems the problem of maximizing a sum of concave convex ratios over a convex set is addressed the projection of the problem onto the image space of the functions that describe the ratios leads to the equivalent problem of maximizing a sum of elementary ratios subject to a linear semi infinite inequality constraint a global optimization algorithm that integrates a branch and bound procedure for dealing with nonconcavities in the image space and an efficient relaxation procedure for handling the semi infinite constraint is proposed and illustrated through numerical examples comparative computational analyses between the proposed algorithm and two alternative algorithms for solving sum of ratios problems are also presented

a practicable branch and bound algorithm for sum of linear ratios problem abstract this article presents a practicable algorithm for globally solving sum of linear ratios problem slr the algorithm works by globally solving a bilinear programming problem eq that is equivalent to the problem slr in the algorithm by utilizing convex envelope and concave envelope of bilinear function the initial nonconvex programming problem is reduced to a sequence of linear relaxation programming problems in order to improve the computational efficiency of the algorithm a new accelerating technique is introduced which provides a theoretical possibility to delete a large part of the investigated region in which there exists no global optimal solution of the eq by combining this innovative technique with branch and bound operations a global optimization algorithm is designed for solving the problem slr finally numerical experimental results show the feasibility and efficiency of the proposed algorithm

determining significant connectivity by 4d spatiotemporal wavelet packet resampling of functional neuroimaging data an active area of neuroimaging research involves examining functional relationships between spatially remote brain regions when determining whether two brain regions exhibit significant correlation due to true functional connectivity one must account for the background spatial correlation inherent in neuroimaging data we define background correlation as spatiotemporal correlation in the data caused by factors other than neurophysiologically based functional associations such as scanner induced correlations and image preprocessing we develop a 4d spatiotemporal wavelet packet resampling method which generates surrogate data that preserves only the average background spatial correlation within an axial slice across axial slices and through each voxel time series while excluding the specific correlations due to true functional relationships we also extend an amplitude adjustment algorithm which adjusts our surrogate data to closely match the amplitude distribution of the original data our method improves upon existing wavelet based methods and extends them to 4d we apply our resampling technique to determine significant functional connectivity from resting state and motor task fmri datasets

dem a variational treatment of dynamic systems this paper presents a variational treatment of dynamic models that furnishes time dependent conditional densities on the path or trajectory of a system s states and the time independent densities of its parameters these are obtained by maximising a variational action with respect to conditional densities under a fixed form assumption about their form the action or path integral of free energy represents a lower bound on the model s log evidence or marginal likelihood required for model selection and averaging this approach rests on formulating the optimisation dynamically in generalised coordinates of motion the resulting scheme can be used for online bayesian inversion of nonlinear dynamic causal models and is shown to outperform existing approaches such as kalman and particle filtering furthermore it provides for dual and triple inferences on a system s states parameters and hyperparameters using exactly the same principles we refer to this approach as dynamic expectation maximisation dem

inferring neural activity from bold signals through nonlinear optimization abstract the blood oxygen level dependent bold fmri signal does not measure neuronal activity directly this fact is a key concern for interpreting functional imaging data based on bold mathematical models describing the path from neural activity to the bold response allow us to numerically solve the inverse problem of estimating the timing and amplitude of the neuronal activity underlying the bold signal in fact these models can be viewed as an advanced substitute for the impulse response function in this work the issue of estimating the dynamics of neuronal activity from the observed bold signal is considered within the framework of optimization problems the model is based on the extended balloon model and describes the conversion of neuronal signals into the bold response through the transitional dynamics of the blood flow inducing signal cerebral blood flow cerebral blood volume and deoxyhemoglobin concentration global optimization techniques are applied to find a control input the neuronal activity and or the biophysical parameters in the model that causes the system to follow an admissible solution to minimize discrepancy between model and experimental data as an alternative to a local linearization ll filtering scheme the optimization method escapes the linearization of the transition system and provides a possibility to search for the global optimum avoiding spurious local minima we have found that the dynamics of the neural signals and the physiological variables as well as the biophysical parameters can be robustly reconstructed from the bold responses furthermore it is shown that spiking off on dynamics of the neural activity is the natural mathematical solution of the model incorporating in addition the expansion of the neural input by smooth basis functions representing a low pass filtering allows us to model local field potential lfp solutions instead of spiking solutions

construction of multivariate surrogate sets from nonlinear data using the wavelet transform abstract the use of surrogate data has become a crucial first step in the study of nonlinearity in time series data a widely used technique to construct surrogate data is to randomize the phases of the data in the fourier domain in this paper an alternative technique based on the resampling of wavelet coefficients is discussed this approach exploits between scale correlations that exist within nonlinear data but which are either absent or weak in stochastic data it proceeds by transforming the data into the wavelet domain and permuting the wavelet coefficients experimental and numerical time series data are used to demonstrate that the performance of the wavelet resampling technique is comparable to phase randomization in terms of the preservation of linear properties removal of nonlinear structure and computational demands however the wavelet technique may have specific and distinct advantages in the application to complex data sets such as numerical analysis of turbulence and experimental brain imaging data where wavelets give a more parsimonious representation of spatio temporal patterns than fourier modes it is shown that different techniques of resampling the data in the wavelet domain may optimize the construction of surrogate data according to the properties of the experimental time series and computational constraints

global optimization of nonlinear sum of ratios problem this article presents a branch and bound algorithm for globally solving the nonlinear sum of ratios problem p on nonconvex feasible region first a problem q is derived which is equivalent to problem p in the algorithm lower bounds are derived by solving a sequence of linear relaxation programming problems which is based on the construction of the linear lower bounding functions for the objective function and the constraint functions of the problem q over the feasible region the proposed branch and bound algorithm is convergent to the global minimum through the successive refinement of the solutions of a series of linear programming problems the numerical experiment is reported to show the feasibility and effectiveness of the proposed algorithm

a wavelet based estimator of the degrees of freedom in denoised fmri time series for probabilistic testing of functional connectivity and brain graphs connectome mapping using techniques such as functional magnetic resonance imaging fmri has become a focus of systems neuroscience there remain many statistical challenges in analysis of functional connectivity and network architecture from bold fmri multivariate time series one key statistic for any time series is its effective degrees of freedom df which will generally be less than the number of time points or nominal degrees of freedom n if we know the df then probabilistic inference on other fmri statistics such as the correlation between two voxel or regional time series is feasible however we currently lack good estimators of df in fmri time series especially after the degrees of freedom of the raw data have been modified substantially by denoising algorithms for head movement here we used a wavelet based method both to denoise fmri data and to estimate the effective df of the denoised process we show that seed voxel correlations corrected for locally variable df could be tested for false positive connectivity with better control over type i error and greater specificity of anatomical mapping than probabilistic connectivity maps using the nominal degrees of freedom we also show that wavelet despiked statistics can be used to estimate all pairwise correlations between a set of regional nodes assign a p value to each edge and then iteratively add edges to the graph in order of increasing p these probabilistically thresholded graphs are likely more robust to regional variation in head movement effects than comparable graphs constructed by thresholding correlations finally we show that time windowed estimates of df can be used for probabilistic connectivity testing or dynamic network analysis so that apparent changes in the functional connectome are appropriately corrected for the effects of transient noise bursts wavelet despiking is both an algorithm for fmri time series denoising and an estimator of the effective df of denoised fmri time series accurate estimation of df offers many potential advantages for probabilistically thresholding functional connectivity and network statistics tested in the context of spatially variant and non stationary noise code for wavelet despiking seed correlational testing and probabilistic graph construction is freely available to download as part of the brainwavelet toolbox at www brainwavelet org

how much information does dependence between wavelet coefficients contain abstractthis article is motivated by several articles that propose statistical inference where the independence of wavelet coefficients for both short as well as long range dependent time series is assumed we focus on the sample variance and investigate the influence of the dependence between wavelet coefficients and this statistic to this end we derive asymptotic distributional properties of the sample variance for a time series that is synthesized ignoring some or all dependence between wavelet coefficients we show that the second order properties differ from the those of the true time series whose wavelet coefficients have the same marginal distribution except in the independent gaussian case this holds true even if the dependency is correct within each level and only the dependence between levels is ignored in the case of sample autocovariances and sample autocorrelations at lag one we indicate that first order properties are erroneous in a second step we investigate several nonparametric b

global optimization for sum of geometric fractional functions this paper presents an efficient branch and bound algorithm for globally solving sum of geometric fractional functions under geometric constraints which arise in various practical problems by using an equivalent transformation and a new linear relaxation technique a linear relaxation programming problem of the equivalent problem is obtained the proposed algorithm is convergent to the global optimal solution by means of the subsequent solutions of a series of linear programming problems numerical results are reported to show the feasibility of our algorithm

testing for nonlinearity in time series the method of surrogate data we describe a statistical approach for identifying nonlinearity in time series the method first specifies some linear process as a null hypothesis then generates surrogate data sets which are consistent with this null hypothesis and finally computes a discriminating statistic for the original and for each of the surrogate data sets if the value computed for the original data is significantly different than the ensemble of values computed for the surrogate data then the null hypothesis is rejected and nonlinearity is detected we discuss various null hypotheses and discriminating statistics the method is demonstrated for numerical data generated by known chaotic systems and applied to a number of experimental time series which arise in the measurement of superfluids brain waves and sunspots we evaluate the statistical significance of the evidence for nonlinear structure in each case and illustrate aspects of the data which this approach identifies

surrogate time series before we apply nonlinear techniques e g those inspired by chaos theory to dynamical phenomena occurring in nature it is necessary to first ask if the use of such advanced techniques is justified by the data while many processes in nature seem very unlikely a priori to be linear the possible nonlinear nature might not be evident in specific aspects of their dynamics the method of surrogate data has become a very popular tool to address such a question however while it was meant to provide a statistically rigorous foolproof framework some limitations and caveats have shown up in its practical use in this paper recent efforts to understand the caveats avoid the pitfalls and to overcome some of the limitations are reviewed and augmented by new material in particular we will discuss specific as well as more general approaches to constrained randomisation providing a full range of examples new algorithms will be introduced for unevenly sampled and multivariate data and for surrogate spike trains the main limitation which lies in the interpretability of the test results will be illustrated through instructive case studies we will also discuss some implementational aspects of the realisation of these methods in the tisean software package

wavelet analysis and synthesis of fractional brownian motion fractional brownian motion fbm offers a convenient modeling for nonstationary stochastic processes with long term dependencies and 1 f type spectral behavior over wide ranges of frequencies statistical self similarity is an essential feature of fbm and makes natural the use of wavelets for both its analysis and its synthesis a detailed second order analysis is carried out for wavelet coefficients of fbm it reveals a stationary structure at each scale and a power law behavior of the coefficients variance from which the fractal dimension of fbm can be estimated conditions for using orthonormal wavelet decompositions as approximate whitening filters are discussed consequences of discretization are considered and some connections between the wavelet point of view and previous approaches based on length measurements analysis or dyadic interpolation synthesis are briefly pointed out

improved surrogate data for nonlinearity tests current tests for nonlinearity compare a time series to the null hypothesis of a gaussian linear stochastic process for this restricted null assumption random surrogates can be constructed which are constrained by the linear properties of the data we propose a more general null hypothesis allowing for nonlinear rescalings of a gaussian linear process we show that such rescalings cannot be accounted for by a simple amplitude adjustment of the surrogates which leads to spurious detection of nonlinearity an iterative algorithm is proposed to make appropriate surrogates which have the same autocorrelations as the data and the same probability distribution

solving sum of quadratic ratios fractional programs via monotonic function in this paper a branch reduce bound algorithm is proposed for globally solving a sum of quadratic ratios fractional programming with nonconvex quadratic constraints due to its intrinsic difficulty less work has been devoted to globally solving this problem the proposed algorithm is based on reformulating the problem as a monotonic optimization problem and it turns out that the optimal solution which is provided by the algorithm is adequately guaranteed to be feasible and to be close to the actual optimal solution convergence of the algorithm is shown and the numerical experiments are given to show the feasibility of the proposed algorithm

constrained linear basis sets for hrf modelling using variational bayes abstract fmri modelling requires flexible haemodynamic response function hrf modelling with the hrf being allowed to vary spatially and between subjects to achieve this flexibility voxelwise parameterised hrfs have been proposed however inference on such models is very slow an alternative approach is to use basis functions allowing inference to proceed in the more manageable general linear model glm framework however a large amount of the subspace spanned by the basis functions produces nonsensical hrf shapes in this work we propose a technique for choosing a basis set and then the means to constrain the subspace spanned by the basis set to only include sensible hrf shapes penny et al neuroimage 2003 showed how variational bayes can be used to infer on the glm for fmri here we extend the work of penny et al to give inference on the glm with constrained hrf basis functions and with spatial markov random fields on the autoregressive noise parameters constraining the subspace spanned by the basis set allows for far superior separation of activating voxels from nonactivating voxels in fmri data we use spatial mixture modelling to produce final probabilities of activation and demonstrate increased sensitivity on an fmri dataset

a model of biological control of plant virus propagation with delays abstract plants are a food source for man and many species they also are sources of medicines fibers for clothes and are essential for a healthy environment but plants are subject to diseases many of these are caused by viruses in plants most of the virus propagation is done by a vector usually insects that bite infected plants and the susceptible plants chemical insecticides are commonly used to control the insects but unfortunately these chemicals have toxic effects on humans animals and the environment in general an alternative is to introduce a new species or just increase the number of a naturally present one to prey on the insects and thus control their number the different steps in the virus propagation process take time in this paper we add a predator to a plant virus propagation model with delays the model is based on delay differential equations there are five populations susceptible and infected plants susceptible and infected vectors and predators the infected vectors have no ill effects from the virus so they do not fight the virus and therefore they do not recover the predators do not get sick from the virus the total number of plants is assumed to be constant since for crops the framers will replace the death plants and one of the plant populations can be determined using the total constant plant population therefore we have a system of five delay ordinary differential equations in five unknowns we assume that predators are introduced at the initial time and after their growth rate has a holling 2 type dependence on the number of vectors we consider two delays the time it takes from the moment a plant is bitten by an infected vector to the moment the plant is infected and a second smaller delay for the time from the moment a vector bites an infected plant to the moment the vector becomes infected we determine the non negative steady state solutions of the system of delay differential equations we analyze their stability by calculating the eigenvalues of the linearized system about each equilibrium point we do numerical simulations for certain values of the parameters finally we search for bifurcation points numerically using the software package biftool conditions are presented for which the disease is controlled or even eliminated from the practical side the model can be used to determine the amount of introduced predators necessary to eliminate the epidemic under different conditions

resampling hierarchical processes in the wavelet domain a case study using atmospheric turbulence abstract there is a growing need for statistical methods that generate an ensemble of plausible realizations of a hierarchical process from a single run or experiment the main challenge is how to construct such an ensemble in a manner that preserves the internal dynamics e g intermittency and temporal persistency a popular hierarchical process often used as a case study in such problems is atmospheric turbulent flow analogies to turbulence are often called upon when information flow from large to small scales non gaussian statistics and intermittency are inherent attributes of the hierarchical process under consideration these attributes are key defining syndromes of the turbulent cascade thereby making turbulence time series ideal for testing such ensemble generation schemes in this study we propose a wavelet based resampling scheme wb and compare it to the traditional fourier based phase randomization bootstrap fb approach within the context of the turbulence energy cascade the comparison between the two resampling methods and observed ensemble statistics constructed by clustering similar meteorological conditions demonstrate that the wb reproduces several features related to intermittency of the ensemble series when compared to fb in particular the wb exhibited an increase in wavelet energy activity and an increase in the wavelet flatness factor with increasing frequency consistent with the cluster of ensemble statistics on the other hand the fb yielded no increase in such energy activity with scale and resulted in near gaussian wavelet coefficients at all frequencies within the inertial subrange the scaling behavior of the longitudinal u and vertical w velocity structure functions of various order p 0 confirms that wb preserves the small scale intermittency whereas fb completely destroys it the extension of wb to the multivariate case is also demonstrated via the conservation of co spectra between u and w time series because the resampling strategy proposed here is conducted in the wavelet domain gap infected and uneven sampled time series can be readily accommodated within the wb finally recommendations about the filter and block sizes are discussed