quantifying temporal correlations a test retest evaluation of functional connectivity in resting state fmri there have been many interpretations of functional connectivity and proposed measures of temporal correlations between bold signals across different brain areas these interpretations yield from many studies on functional connectivity using resting state fmri data that have emerged in recent years however not all of these studies used the same metrics for quantifying the temporal correlations between brain regions in this paper we use a public domain test retest resting state fmri data set to perform a systematic investigation of the stability of the metrics that are often used in resting state functional connectivity fc studies the fmri data set was collected across three different sessions the second session took place approximately eleven months after the first session and the third session was an hour after the second session the fc metrics composed of cross correlation partial cross correlation cross coherence and parameters based on an autoregressive model we discussed the strengths and weaknesses of each metric we performed roi level and full brain seed based voxelwise test retest analyses using each fc metric to assess its stability for both roi level and voxel level analyses we found that cross correlation yielded more stable measurements than the other metrics we discussed the consequences of this result on the utility of the fc metrics we observed that for negatively correlated rois their partial cross correlation is shrunk towards zero thus affecting the stability of their fc for the present data set we found greater stability in fc between the second and third sessions one hour between sessions compared to the first and second sessions approximately 11 months between sessions finally we report that some of the metrics showed a positive association between strength and stability in summary the results presented in this paper suggest important implications when choosing metrics for quantifying and assessing various types of functional connectivity for resting state fmri studies

short and long run causality measures theory and inference abstract the concept of causality introduced by wiener wiener n 1956 the theory of prediction in e f beckenback ed the theory of prediction mcgraw hill new york chapter 8 and granger granger c w j 1969 investigating causal relations by econometric models and cross spectral methods econometrica 37 424 459 is defined in terms of predictability one period ahead this concept can be generalized by considering causality at any given horizon h as well as tests for the corresponding non causality dufour j m renault e 1998 short run and long run causality in time series theory econometrica 66 1099 1125 dufour j m pelletier d renault e 2006 short run and long run causality in time series inference journal of econometrics 132 2 337 362 instead of tests for non causality at a given horizon we study the problem of measuring causality between two vector processes existing causality measures have been defined only for the horizon 1 and they fail to capture indirect causality we propose generalizations to any horizon h of the measures introduced by geweke geweke j 1982 measurement of linear dependence and feedback between multiple time series journal of the american statistical association 77 304 313 nonparametric and parametric measures of unidirectional causality and instantaneous effects are considered on noting that the causality measures typically involve complex functions of model parameters in var and varma models we propose a simple simulation based method to evaluate these measures for any varma model we also describe asymptotically valid nonparametric confidence intervals based on a bootstrap technique finally the proposed measures are applied to study causality relations at different horizons between macroeconomic monetary and financial variables in the us

a copula approach to assessing granger causality in neuroscience as in many other fields of science and engineering it is crucial to assess the causal interactions among multivariate time series granger causality has been increasingly used to identify causal influence between time series based on multivariate autoregressive models such an approach is based on linear regression framework with implicit gaussian assumption of model noise residuals having constant variance as a consequence this measure cannot detect the cause effect relationship in high order moments and nonlinear causality here we propose an effective model free copula based granger causality measure that can be used to reveal nonlinear and high order moment causality we first formulate granger causality as the log likelihood ratio in terms of conditional distribution and then derive an efficient estimation procedure using conditional copula we use resampling techniques to build a baseline null hypothesis distribution from which statistical significance can be derived we perform a series of simulations to investigate the performance of our copula based granger causality and compare its performance against other state of the art techniques our method is finally applied to neural field potential time series recorded from visual cortex of a monkey while performing a visual illusion task

a study of problems encountered in granger causality analysis from a neuroscience perspective granger causality methods were developed to analyze the flow of information between time series these methods have become more widely applied in neuroscience frequency domain causality measures such as those of geweke as well as multivariate methods have particular appeal in neuroscience due to the prevalence of oscillatory phenomena and highly multivariate experimental recordings despite its widespread application in many fields there are ongoing concerns regarding the applicability of granger causality methods in neuroscience when are these methods appropriate how reliably do they recover the system structure underlying the observed data what do frequency domain causality measures tell us about the functional properties of oscillatory neural systems in this paper we analyze fundamental properties of granger geweke gg causality both computational and conceptual specifically we show that i gg causality estimates can be either severely biased or of high variance both leading to spurious results ii even if estimated correctly gg causality estimates alone are not interpretable without examining the component behaviors of the system model and iii gg causality ignores critical components of a system s dynamics based on this analysis we find that the notion of causality quantified is incompatible with the objectives of many neuroscience investigations leading to highly counterintuitive and potentially misleading results through the analysis of these problems we provide important conceptual clarification of gg causality with implications for other related causality approaches and for the role of causality analyses in neuroscience as a whole

an approximate generalized linear model with random effects for informative missing data this paper develops a class of models to deal with missing data from longitudinal studies we assume that separate models for the primary response and missingness e g number of missed visits are linked by a common random parameter such models have been developed in the econometrics heckman 1979 econometrica 47 153 161 and biostatistics wu and carroll 1988 biometrics 44 175 188 literature for a gaussian primary response we allow the primary response conditional on the random parameter to follow a generalized linear model and approximate the generalized linear model by conditioning on the data that describes missingness the resultant approximation is a mixed generalized linear model with possibly heterogeneous random effects an example is given to illustrate the approximate approach and simulations are performed to critique the adequacy of the approximation for repeated binary data

granger geweke causality estimation and interpretation abstract in a recent pnas article1 stokes and purdon performed numerical simulations to argue that granger geweke causality ggc estimation is severely biased or of high variance and ggc application to neuroscience is problematic because the ggc measure is independent of receiver dynamics here we use the same simulation examples to show that ggc measures when properly estimated either via the spectral factorization enabled nonparametric approach or the var model based parametric approach do not have the claimed bias and high variance problems further the receiver independence property of ggc does not present a problem for neuroscience applications when the nature and context of experimental measurements are taken into consideration ggc along with other spectral quantities yield neurophysiologically interpretable results

some recent development in a concept of causality abstract the paper considers three separate but related topics i what is the relationship between causation and co integration if a pair of i 1 series are co integration there must be causation in at least one direction an implication is that some tests of causation based on different series may have missed one source of causation ii is there a need for a definition of instantaneous causation in a decision science it is argued that no such definition is required iii can causality tests be used for policy evaluation it is suggested that these tests are useful but that they should be evaluated with care

financial networks based on granger causality a case study connectivity analysis is performed on a long financial record of 21 international stock indices employing a linear and a nonlinear causality measure the conditional granger causality index cgci and the partial mutual information on mixed embedding pmime respectively both measures aim to specify the direction of the interrelationships among the international stock indexes and portray the links of the resulting networks by the presence of direct couplings between variables exploiting all available information however their differences are assessed due to the presence of nonlinearity the weighted networks formed with respect to the causality measures are transformed to binary ones using a significance test the financial networks are formed on sliding windows in order to examine the network characteristics and trace changes in the connectivity structure subsequently two statistical network quantities are calculated the average degree and the average shortest path length the empirical findings reveal interesting time varying properties of the constructed network which are clearly dependent on the nature of the financial cycle

causality analysis of neural connectivity critical examination of existing methods and advances of new methods granger causality gc is one of the most popular measures to reveal causality influence of time series and has been widely applied in economics and neuroscience especially its counterpart in frequency domain spectral gc as well as other granger like causality measures have recently been applied to study causal interactions between brain areas in different frequency ranges during cognitive and perceptual tasks in this paper we show that 1 gc in time domain cannot correctly determine how strongly one time series influences the other when there is directional causality between two time series and 2 spectral gc and other granger like causality measures have inherent shortcomings and or limitations because of the use of the transfer function or its inverse matrix and partial information of the linear regression model on the other hand we propose two novel causality measures in time and frequency domains for the linear regression model called new causality and new spectral causality respectively which are more reasonable and understandable than gc or granger like measures especially from one simple example we point out that in time domain both new causality and gc adopt the concept of proportion but they are defined on two different equations where one equation for gc is only part of the other for new causality thus the new causality is a natural extension of gc and has a sound conceptual theoretical basis and gc is not the desired causal influence at all by several examples we confirm that new causality measures have distinct advantages over gc or granger like measures finally we conduct event related potential causality analysis for a subject with intracranial depth electrodes undergoing evaluation for epilepsy surgery and show that in the frequency domain all measures reveal significant directional event related causality but the result from new spectral causality is consistent with event related time frequency power spectrum activity the spectral gc as well as other granger like measures are shown to generate misleading results the proposed new causality measures may have wide potential applications in economics and neuroscience

the aic criterion and symmetrizing the kullback leibler divergence the akaike information criterion aic is a widely used tool for model selection aic is derived as an asymptotically unbiased estimator of a function used for ranking candidate models which is a variant of the kullback leibler divergence between the true model and the approximating candidate model despite the kullback leibler s computational and theoretical advantages what can become inconvenient in model selection applications is their lack of symmetry simple examples can show that reversing the role of the arguments in the kullback leibler divergence can yield substantially different results in this paper three new functions for ranking candidate models are proposed these functions are constructed by symmetrizing the kullback leibler divergence between the true model and the approximating candidate model the operations used for symmetrizing are the average geometric and harmonic means it is found that the original aic criterion is an asymptotically unbiased estimator of these three different functions using one of these proposed ranking functions an example of new bias correction to aic is derived for univariate linear regression models a simulation study based on polynomial regression is provided to compare the different proposed ranking functions with aic and the new derived correction with aicc

shortcomings limitations of blockwise granger causality and advances of blockwise new causality multivariate blockwise granger causality bgc is used to reflect causal interactions among blocks of multivariate time series in particular spectral bgc and conditional spectral bgc are used to disclose blockwise causal flow among different brain areas in various frequencies in this paper we demonstrate that 1 bgc in time domain may not necessarily disclose true causality and 2 due to the use of the transfer function or its inverse matrix and partial information of the multivariate linear regression model both of spectral bgc and conditional spectral bgc have shortcomings and or limitations which may inevitably lead to misinterpretation we then in time and frequency domains develop two new multivariate blockwise causality methods for the linear regression model called blockwise new causality bnc and spectral bnc respectively by several examples we confirm that bnc measures are more reasonable and sensitive to reflect true causality or trend of true causality than bgc or conditional bgc finally for electroencephalograph data from an epilepsy patient we analyze event related potential causality and demonstrate that both of the bgc and bnc methods show significant causality flow in frequency domain but the spectral bnc method yields satisfactory and convincing results which are consistent with an event related time frequency power spectrum activity the spectral bgc method is shown to generate misleading results thus we deeply believe that our new blockwise causality definitions as well as our previous nc definitions may have wide applications to reflect true causality among two blocks of time series or two univariate time series in economics neuroscience and engineering

measures of mutual and causal dependence between two time series corresp new measures are proposed for mutual and causal dependence between two time series based on information theoretical ideas the measure of mutual dependence is shown to be the sum of the measure of unidirectional causal dependence from the first time series to the second the measure of unidirectional causal dependence from the second to the first and the measure of instantaneous causal dependence the measures are applicable to any kind of time series continuous discrete or categorical

framework to study dynamic dependencies in networks of interacting processes the analysis of dynamic dependencies in complex systems such as the brain helps to understand how emerging properties arise from interactions here we propose an information theoretic framework to analyze the dynamic dependencies in multivariate time evolving systems this framework constitutes a fully multivariate extension and unification of previous approaches based on bivariate or conditional mutual information and granger causality or transfer entropy we define multi information measures that allow us to study the global statistical structure of the system as a whole the total dependence between subsystems and the temporal statistical structure of each subsystem we develop a stationary and a nonstationary formulation of the framework we then examine different decompositions of these multi information measures the transfer entropy naturally appears as a term in some of these decompositions this allows us to examine its properties not as an isolated measure of interdependence but in the context of the complete framework more generally we use causal graphs to study the specificity and sensitivity of all the measures appearing in these decompositions to different sources of statistical dependence arising from the causal connections between the subsystems we illustrate that there is no straightforward relation between the strength of specific connections and specific terms in the decompositions furthermore causal and noncausal statistical dependencies are not separable in particular the transfer entropy can be nonmonotonic in dependence on the connectivity strength between subsystems and is also sensitive to internal changes of the subsystems so it should not be interpreted as a measure of connectivity strength altogether in comparison to an analysis based on single isolated measures of interdependence this framework is more powerful to analyze emergent properties in multivariate systems and to characterize functionally relevant changes in the dynamics

state space analysis of granger geweke causality measures with application to fmri the recent interest in the dynamics of networks and the advent across a range of applications of measuring modalities that operate on different temporal scales have put the spotlight on some significant gaps in the theory of multivariate time series fundamental to the description of network dynamics is the direction of interaction between nodes accompanied by a measure of the strength of such interactions granger causality and its associated frequency domain strength measures gems due to geweke provide a framework for the formulation and analysis of these issues in pursuing this setup three significant unresolved issues emerge first computing gems involves computing submodels of vector time series models for which reliable methods do not exist second the impact of filtering on gems has never been definitively established third the impact of downsampling on gems has never been established in this work using state space methods we resolve all these issues and illustrate the results with some simulations our analysis is motivated by some problems in fmri brain imaging to which we apply it but it is of general applicability

functional connectivity studying nonlinear delayed interactions between bold signals correlation analysis has been widely used in the study of functional connectivity based on fmri data it assumes that the relevant information about the interactions of brain regions is reflected by a linear relationship between the values of two signals at the same time however this hypothesis has not been thoroughly investigated yet in this work we study in depth the information shared by bold signals of pairs of brain regions in particular we assess the amount of nonlinear and or nonsynchronous interactions present in data this is achieved by testing models reflecting linear synchronous interactions against more general models encompassing nonlinear nonsynchronous interactions many factors influencing measured bold signals are critical for the study of connectivity such as paradigm induced bold responses preprocessing motion artifacts and geometrical distortions interactions are also influenced by the proximity of brain regions the influence of all these factors is taken into account and the nature of the interactions is studied using various experimental conditions such that the conclusions reached are robust with respect to variation of these factors after defining nonlinear and or nonsynchronous interaction models in the framework of general linear models statistical tests are performed on different fmri data sets to infer the nature of the interactions finally a new connectivity metric is proposed which takes these inferences into account we find that bold signal interactions are statistically more significant when taking into account the history of the distant signal i e the signal from the interacting region than when using a model of linear instantaneous interaction moreover about 75 of the interactions are symmetric as assessed with the proposed connectivity metric the history dependent part of the coupling between brain regions can explain a high percentage of the variance in the data sets studied as these results are robust with respect to various confounding factors this work suggests that models used to study the functional connectivity between brain areas should in general take the bold signal history into account

measures of conditional linear dependence and feedback between time series abstract measures of linear dependence and feedback for two multiple time series conditional on a third are defined the measure of conditional linear dependence is the sum of linear feedback from the first to the second conditional on the third linear feedback from the second to the first conditional on the third and instantaneous linear feedback between the first and second series conditional on the third the measures are non negative and may be expressed in terms of measures of unconditional feedback between various combinations of the three series the measures of conditional linear feedback can be additively decomposed by frequency estimates of these measures are straightforward to compute and their distribution can be routinely approximated by bootstrap methods an empirical example involving real output money and interest rates is presented

robust causality test of infinite variance processes abstract this paper develops a robust causality test for time series with infinite variance innovation processes first we introduce a measure of dependence for vector nonparametric linear processes and derive the asymptotic distribution of the test statistic by taniguchi et al 1996 in the infinite variance case second we construct a weighted version of the generalized empirical likelihood gel test statistic called the self weighted gel statistic in the time domain the limiting distribution of the self weighted gel test statistic is shown to be the usual chi squared one regardless of whether the model has finite variance or not some simulation experiments illustrate satisfactory finite sample performances of the proposed test

disclosing large scale directed functional connections in meg with the multivariate phase slope index abstract the phase slope index psi is a method to disclose the direction of frequency specific neural interactions from magnetoencephalographic meg time series a fundamental property of psi is that of vanishing for linear mixing of independent neural sources this property allows psi to cope with the artificial instantaneous connectivity among meg sensors or brain sources induced by the field spread nevertheless psi is limited by being a bivariate estimator of directionality as opposite to the multidimensional nature of brain activity as revealed by meg the purpose of this work is to provide a multivariate generalization of psi we termed this measure as the multivariate phase slope index mpsi in order to test the ability of mpsi in estimating the directionality and to compare the mpsi results to those obtained by bivariate psi approaches based on maximizing imaginary part of coherency and on canonical correlation analysis we used extensive simulations we proved that mpsi achieves the highest performance and that in a large number of simulated cases the bivariate methods as opposed to mpsi do not detect a statistically significant directionality finally we applied mpsi to assess seed based directed functional connectivity in the alpha band from resting state meg data of 61 subjects from the human connectome project the obtained results highlight a directed functional coupling in the alpha band between the primary visual cortex and several key regions of well known resting state networks e g dorsal attention network and fronto parietal network

norm estimates of complex symmetric operators applied to quantum systems this paper communicates recent results in the theory of complex symmetric operators and shows through two non trivial examples their potential usefulness in the study of schrodinger operators in particular we propose a formula for computing the norm of a compact complex symmetric operator this observation is applied to two concrete problems related to quantum mechanical systems first we give sharp estimates on the exponential decay of the resolvent and the single particle density matrix for schrodinger operators with spectral gaps second we provide new ways of evaluating the resolvent norm for schrodinger operators appearing in the complex scaling theory of resonances

unified framework for information integration based on information geometry assessment of causal influences is a ubiquitous and important subject across diverse research fields drawn from consciousness studies integrated information is a measure that defines integration as the degree of causal influences among elements whereas pairwise causal influences between elements can be quantified with existing methods quantifying multiple influences among many elements poses two major mathematical difficulties first overestimation occurs due to interdependence among influences if each influence is separately quantified in a part based manner and then simply summed over second it is difficult to isolate causal influences while avoiding noncausal confounding influences to resolve these difficulties we propose a theoretical framework based on information geometry for the quantification of multiple causal influences with a holistic approach we derive a measure of integrated information which is geometrically interpreted as the divergence between the actual probability distribution of a system and an approximated probability distribution where causal influences among elements are statistically disconnected this framework provides intuitive geometric interpretations harmonizing various information theoretic measures in a unified manner including mutual information transfer entropy stochastic interaction and integrated information each of which is characterized by how causal influences are disconnected in addition to the mathematical assessment of consciousness our framework should help to analyze causal relationships in complex systems in a complete and hierarchical manner

small sample inference for incomplete longitudinal data with truncation and censoring in tumor xenograft models in cancer drug development demonstrating activity in xenograft models where mice are grafted with human cancer cells is an important step in bringing a promising compound to humans a key outcome variable is the tumor volume measured in a given period of time for groups of mice given different doses of a single or combination anticancer regimen however a mouse may die before the end of a study or may be sacrificed when its tumor volume quadruples and its tumor may be suppressed for some time and then grow back thus incomplete repeated measurements arise the incompleteness or missingness is also caused by drastic tumor shrinkage 0 01 cm3 or random truncation because of the small sample sizes in these models asymptotic inferences are usually not appropriate we propose two parametric test procedures based on the em algorithm and the bayesian method to compare treatment effects among different groups while accounting for informative censoring a real xenograft study on a new antitumor agent temozolomide combined with irinotecan is analyzed using the proposed methods

directionality of coupling from bivariate time series how to avoid false causalities and missed connections we discuss some problems encountered in inference of directionality of coupling or in the case of two interacting systems in inference of causality from bivariate time series we identify factors and influences that can lead to either decreased test sensitivity or false detections and propose ways to cope with them in order to perform tests with high sensitivity and a low rate of false positive results

assessing granger non causality using nonparametric measure of conditional independence in recent years granger causality has become a popular method in a variety of research areas including engineering neuroscience and economics however despite its simplicity and wide applicability the linear granger causality is an insufficient tool for analyzing exotic stochastic processes such as processes involving non linear dynamics or processes involving causality in higher order statistics in order to analyze such processes more reliably a different approach toward granger causality has become increasingly popular this new approach employs conditional independence as a tool to discover granger non causality without any assumption on the underlying stochastic process this paper discusses the concept of discovering granger non causality using measures of conditional independence and proposes a novel measure of conditional independence in brief the proposed approach estimates the conditional distribution function through a kernel based least square regression approach this paper also explores the strengths and weaknesses of the proposed method compared to other available methods and provides a detailed comparison of these methods using a variety of synthetic data sets

transfer entropy as a log likelihood ratio transfer entropy an information theoretic measure of time directed information transfer between joint processes has steadily gained popularity in the analysis of complex stochastic dynamics in diverse fields including the neurosciences ecology climatology and econometrics we show that for a broad class of predictive models the log likelihood ratio test statistic for the null hypothesis of zero transfer entropy is a consistent estimator for the transfer entropy itself for finite markov chains furthermore no explicit model is required in the general case an asymptotic 2 distribution is established for the transfer entropy estimator the result generalizes the equivalence in the gaussian case of transfer entropy and granger causality a statistical notion of causal influence based on prediction via vector autoregression and establishes a fundamental connection between directed information transfer and causality in the wiener granger sense

analysis of economic growth fluctuations based on eemd and causal decomposition abstract in this paper we first apply ensemble empirical mode decomposition eemd to analyze the change rate of gdp time series for ten major countries each signal is decomposed into four modes with different scales and a residual trend the variance contribution and averaged period cycle for each mode is then calculated results show that economic growth fluctuations for most of the countries mainly fluctuate at a short period between 3 5 years then the causal decomposition proposed by yang et al is utilized to detect the mutual causation among different countries the novelty of this method is that the causal interaction is identified in instantaneous phase coherence at a specific time scale the strengths and directions of causal relationship for each mode among countries differ from each other which mirrors the fact that the world economic activities are fluctuated and changeable over varying periods however no visible causation for the fourth mode long term cycle among countries is characterized meaning that the development of economy over a long period for most countries depends on its own conditions

granger causality maps of diffusion processes granger causality is a statistical concept devised to reconstruct and quantify predictive information flow between stochastic processes although the general concept can be formulated model free it is often considered in the framework of linear stochastic processes here we show how local linear model descriptions can be employed to extend granger causality into the realm of nonlinear systems this novel treatment results in maps that resolve granger causality in regions of state space through examples we provide a proof of concept and illustrate the utility of these maps moreover by integration we convert the local granger causality into a global measure that yields a consistent picture for a global ornstein uhlenbeck process finally we recover invariance transformations known from the theory of autoregressive processes

analysis of multi arm tumor growth trials in xenograft animals using phase change adaptive piecewise quadratic models xenograft trials allow tumor growth in human cell lines to be monitored over time in a mouse model we consider the problem of inferring the effect of treatment combinations on tumor growth a piecewise quadratic model with flexible phase change locations is proposed to model the effect of change in therapy over time each piece represents a growth phase with phase changes in response to change in treatment piecewise slopes represent phase specific log linear growth rates and curvature parameters represent departure from linear growth trial data are analyzed in two stages i subject specific curve fitting ii analysis of slope and curvature estimates across subjects a least squares approach with penalty for phase change point location is proposed for curve fitting in simulation studies the method is shown to give consistent estimates of slope and curvature parameters under independent and ar 1 measurement error the piecewise quadratic model is shown to give excellent fit median r 2 0 98 to growth data from a six armed xenograft trial on a lung carcinoma cell line

regression and time series model selection in small samples summary a bias correction to the akaike information criterion aic is derived for regression and autoregressive time series models the correction is of particular use when the sample size is small or when the number of fitted parameters is a moderate to large fraction of the sample size the corrected method called aicc is asymptotically efficient if the true model is infinite dimensional furthermore when the true model is of finite dimension aicc is found to provide better model order choices than any other asymptotically efficient method applications to nonstationary autoregressive and mixed autoregressive moving average time series models are also discussed

entropy and predictability of stock market returns abstract we examine the predictability of stock market returns by employing a new metric entropy measure of dependence with several desirable properties we compare our results with a number of traditional measures the metric entropy is capable of detecting nonlinear dependence within the returns series and is also capable of detecting nonlinear affinity between the returns and their predictions obtained from various models thereby serving as a measure of out of sample goodness of fit or model adequacy several models are investigated including the linear and neural network models as well as nonparametric and recursive unconditional mean models we find significant evidence of small nonlinear unconditional serial dependence within the returns series but fragile evidence of superior conditional predictability profit opportunity when using market switching versus buy and hold strategies

how to detect the granger causal flow direction in the presence of additive noise granger causality metrics have become increasingly popular tools to identify directed interactions between brain areas however it is known that additive noise can strongly affect granger causality metrics which can lead to spurious conclusions about neuronal interactions to solve this problem previous studies have proposed the detection of granger causal directionality i e the dominant granger causal flow using either the slope of the coherency phase slope index psi or by comparing granger causality values between original and time reversed signals reversed granger testing we show that for ensembles of vector autoregressive var models encompassing bidirectionally coupled sources these alternative methods do not correctly measure granger causal directionality for a substantial fraction of var models even in the absence of noise we then demonstrate that uncorrelated noise has fundamentally different effects on directed connectivity metrics than linearly mixed noise where the latter may result as a consequence of electric volume conduction uncorrelated noise only weakly affects the detection of granger causal directionality whereas linearly mixed noise causes a large fraction of false positives for standard granger causality metrics and psi but not for reversed granger testing we further show that we can reliably identify cases where linearly mixed noise causes a large fraction of false positives by examining the magnitude of the instantaneous influence coefficient in a structural var model by rejecting cases with strong instantaneous influence we obtain an improved detection of granger causal flow between neuronal sources in the presence of additive noise these techniques are applicable to real data which we demonstrate using actual area v1 and area v4 lfp data recorded from the awake monkey performing a visual attention task

kernel granger causality mapping effective connectivity on fmri data although it is accepted that linear granger causality can reveal effective connectivity in functional magnetic resonance imaging fmri the issue of detecting nonlinear connectivity has hitherto not been considered in this paper we address kernel granger causality kgc to describe effective connectivity in simulation studies and real fmri data of a motor imagery task based on the theory of reproducing kernel hilbert spaces kgc performs linear granger causality in the feature space of suitable kernel functions assuming an arbitrary degree of nonlinearity our results demonstrate that kgc captures effective couplings not revealed by the linear case in addition effective connectivity networks between the supplementary motor area sma as the seed and other brain areas are obtained from kgc

a connectivity based method for defining regions of interest in fmri data in this paper we describe a new methodology for defining brain regions of interset rois in functional magnetic resonance imaging fmri data the rois are defined based on their functional connectivity to other rois i e rois are defined as sets of voxels with similar connectivity patterns to other rois the method relies on 1 a spatially regularized canonical correlation analysis for identifying maximally correlated signals which are not due to correlated noise 2 a test for merging rois which have similar connectivity patterns to the other rois and 3 a graph cuts optimization for assigning voxels to rois since our method is fully connectivity based the extracted rois and their corresponding time signals are ideally suited for a subsequent brain connectivity analysis

evaluating frequency wise directed connectivity of bold signals applying relative power contribution with the linear multivariate time series models in this article we propose a statistical method to evaluate directed interactions of functional magnetic resonance imaging fmri data the multivariate autoregressive mar model was combined with the relative power contribution rpc in this analysis the mar model was fitted to the data to specify the direction of connections and the rpc quantifies the strength of connections as the rpc is computed in the frequency domain we can evaluate the connectivity for each frequency component from this we can establish whether the specified connections represent low or high frequency connectivity which cannot be examined solely using the estimated mar coefficients we applied this analysis method to fmri data obtained during visual motion tasks confirming previous reports of bottom up connectivity around the frequency corresponding to the block experimental design furthermore we used the mar model with exogenous variables marx to extend our understanding of these data and to show how the input to v1 transfers to higher cortical areas

measurement of linear dependence and feedback between multiple time series abstract measures of linear dependence and feedback for multiple time series are defined the measure of linear dependence is the sum of the measure of linear feedback from the first series to the second linear feedback from the second to the first and instantaneous linear feedback the measures are nonnegative and zero only when feedback causality of the relevant type is absent the measures of linear feedback from one series to another can be additively decomposed by frequency a readily usable theory of inference for all of these measures and their decompositions is described the computations involved are modest

conditions for bound states in a periodic linear chain and the spectra of a class of toeplitz operators in terms of polylogarithm functions conditions for bound states for a periodic linear chain are given within the framework of an exactly solvable non relativistic quantum mechanical model in three dimensional space these conditions express the strength parameter in terms of the distance between two consecutive centres of the chain and of the range interaction parameter this expression can be formulated in terms of polylogarithm functions and in some particular cases in terms of the riemann zeta function an interesting mathematical result is that these expressions also correspond to the spectra of toeplitz complex symmetric operators the non trivial zeros of the riemann zeta function are interpreted as multiple points at the origin of the spectra of these toeplitz operators

temporal aggregation of multivariate garch processes this paper derives results for the temporal aggregation of multivariate garch 1 1 processes in the general vector specification it is shown that the class of weak multivariate garch 1 1 processes is closed under temporal aggregation fourth moment characteristics turn out to be crucial for the low frequency dynamics for both stock and flow variables in some aspects the aggregation characteristics of multivariate garch processes are shown to be different from those of vector autoregressive moving average processes a numerical example illustrates some of the results

time evolution of wave packets on nanostructures time evolution of wave packets on nanostructures is studied on the basis of a three dimensional solvable model with singular interactions de prunele 1997 j phys a math gen 30 7831 in particular methods and tools are provided to determine time independent upper bounds for the overlap of the normalized time dependent wave packet with the time independent normalized wave packet concentrated at an arbitrarily chosen vertex of the nanosystem the set of upper bounds referring to all initial positions of the wave packet and all overlaps are summarized in a matrix the analytical formulation allows a detailed study for arbitrary geometrical configurations time evolution on truncated quasicrystalline systems has been found to be site selective depending on the position of the initial wave packet

generalized norton simon models of tumour growth this paper considers the analysis of serial data on the growth of tumours in laboratory rodents i propose a model a generalization of the tumour growth model of norton and simon that leads to a rich family of growth and decay curves the model assumes that unperturbed growth follows the generalized logistic form it accommodates time varying treatment effects through an effective dose function i fit two such models to data on a human prostate tumour growing in nude mice and compare the fitted curves and dose functions with a non parametric curve and dose function estimated from a cubic spline model all three models account for both random animal effects and autocorrelation monte carlo results suggest that a maximum likelihood estimates of growth parameters are biased although not severely and b standard errors are conservative in small samples but become increasingly accurate in larger samples

solvable quantum mechanical model in two dimensional space a one particle non relativistic quantum mechanical solvable model in two dimensional space is given the hamiltonian is the sum of kinetic and interaction parts interactions are separable and can be centred at n arbitrary points of the plane conditions for the existence and for the number of bound states in finite linear chains are formulated in terms of the parameters of the interactions and intercentre distances scattering problems are also considered finally when the interactions are centred in a single centre it is shown that the model remains solvable in the presence of a uniform magnetic field of arbitrary intensity

nonparametric estimation and inference for conditional density based granger causality measures we propose a nonparametric estimation and inference for conditional density based granger causality measures that quantify linear and nonlinear granger causalities we first show how to write the causality measures in terms of copula densities thereafter we suggest consistent estimators for these measures based on a consistent nonparametric estimator of copula densities furthermore we establish the asymptotic normality of these nonparametric estimators and discuss the validity of a local smoothed bootstrap that we use in finite sample settings to compute a bootstrap bias corrected estimator and to perform statistical tests a monte carlo simulation study reveals that the bootstrap bias corrected estimator behaves well and the corresponding test has quite good finite sample size and power properties for a variety of typical data generating processes and different sample sizes finally two empirical applications are considered to illustrate the practical relevance of nonparametric causality measures

granger causality analysis with nonuniform sampling and its application to pulse coupled nonlinear dynamics the granger causality gc analysis is an effective approach to infer causal relations for time series however for data obtained by uniform sampling i e with an equal sampling time interval it is known that gc can yield unreliable causal inference due to aliasing if the sampling rate is not sufficiently high to solve this unreliability issue we consider the nonuniform sampling scheme as it can mitigate against aliasing by developing an unbiased estimation of power spectral density of nonuniformly sampled time series we establish a framework of spectrum based nonparametric gc analysis applying this framework to a general class of pulse coupled nonlinear networks and utilizing some particular spectral structure possessed by these nonlinear network data we demonstrate that for such nonlinear networks with nonuniformly sampled data reliable gc inference can be achieved at a low nonuniform mean sampling rate at which the traditional uniform sampling gc may lead to spurious causal inference

the effect of filtering on granger causality based multivariate causality measures in the past causality measures based on granger causality have been suggested for assessing directionality in neural signals in frequency domain analyses power or coherence of neural data it is common to preprocess the time series by filtering or decimating however in other fields it has been shown theoretically that filtering in combination with granger causality may lead to spurious or missed causalities we investigated whether this result translates to multivariate causality methods derived from granger causality with a a simulation study and b an application to magnetoencephalographic data to this end we performed extensive simulations of the effect of applying different filtering techniques and evaluated the performance of five different multivariate causality measures in combination with two numerical significance measures random permutation and leave one out method the analysis included three of the most widely used filters high pass low pass notch filter four different filter types butterworth chebyshev i and ii elliptic filter variation of filter order decimating and interpolation the simulation results suggest that preprocessing without a strong prior about the artifact to be removed disturbs the information content and time ordering of the data and leads to spurious and missed causalities only if apparent artifacts like a current or movement artifact are present filtering out the respective disturbance seems advisable while oversampling poses no problem decimation by a factor greater than the minimum time shift between the time series may lead to wrong inferences in general the multivariate causality measures are very sensitive to data preprocessing

a neural network model of causality this paper proposes a model for commonsense causal reasoning based on the basic idea of neural networks after an analysis of the advantages and limitations of existing accounts of causality a fuzzy logic based formalism fel is proposed that takes into account the inexactness and the cumulative evidentiality of commonsense causal reasoning overcoming the limitations of existing accounts analyses concerning how fel handles various aspects of commonsense causal reasoning are performed in an abstract way fel can be implemented naturally in a neural connectionist network this work also serves to link rule based reasoning with neural network models in that a rule encoding scheme fel is equated directly to a neural network model

connectivity analysis with structural equation modelling an example of the effects of voxel selection abstract structural equation modelling sem of neuroimaging data is commonly applied to a network of distributed brain regions we applied sem to an fmri dataset to identify condition specific effects in a simple experiment composed of visual stimulation and baseline conditions the visual network was composed of three well defined anatomical regions v1 v2 and v5 and three path connections v1 v2 v1 v5 and v2 v5 this network was used to test four hypotheses 1 whether the condition specific effects for all three connections vary according to the data selected for modelling 2 whether the summary measures that are often used are indeed appropriate 3 whether measures taken from the voxel timecourse can reliably predict the condition specific effects for each one of the three path connections and 4 whether all voxels within an anatomical region yield equivalent sem outcomes there was some variability in the significance of the condition specific effects across randomly selected voxels within regions however the sem outcome from the summary measures was comparable to the most frequent pattern of condition specific effects magnitude delay spread and goodness of fit measures taken from a gamma fit to the voxel time courses predicted reliably the significance of the sem condition specific effects for each connection this result enabled us to identify spatially coherent regions at the boundaries of v2 that displayed different condition specific effects from those seen in the majority of the voxels although the generality of these results awaits further investigation this example highlights a number of important issues for sem we have provided further evidence that the sem outcome does vary somewhat according to the voxels selected and that although the use of summary measures can give a generalised view of the connectivity pattern they could fail to capture functional differences within specialised areas

causal conditioning and instantaneous coupling in causality graphs this study aims at providing the definitive links between massey and kramer s directed information theory and granger causality graphs recently formalized by eichler this naturally leads to consider two concepts of causality that can occur in physical systems dynamical causality and instantaneous coupling although it is well accepted that the former is assessed by conditional transfer entropy the latter is often overlooked even if it was clearly introduced and understood in seminal studies in the bivariate case we show that directed information is decomposed into the sum of transfer entropy and instantaneous information exchange in the multivariate case encountered for conditional graph modeling such a decomposition does not hold anymore we provide a new insight into the problem of instantaneous coupling and show how it affects the estimated structure of a graphical model that should provide a sparse representation of a complex system this result is discussed and analyzed for the inference of causality graphs two synthetic examples are developed to illustrate the theoretical concepts practical issues are also briefly discussed on these examples and an extension of leonenko s k nearest neighbors based entropy estimators is used to derive a nonparametric estimator of directed information

source coding with feed forward rate distortion theorems and error exponents for a general source in this work we consider a source coding model with feed forward we analyze a system with a noiseless feed forward link where the decoder has knowledge of all previous source samples while reconstructing the present sample the rate distortion function for an arbitrary source with feed forward is derived in terms of directed information a variant of mutual information we further investigate the nature of the rate distortion function with feed forward for two common types of sources discrete memory less sources and gaussian sources we then characterize the error exponent for a general source with feed forward the results are then extended to feed forward with an arbitrary delay larger than the block length

inference on one way effect and evidence in japanese macroeconomic data this paper provides an approach to testing a variety of causal characteristics expressed in terms of the measures of one way effect for cointegrated vector time series for this purpose we propose wald tests and their computational algorithm by means of the measures of one way effect incorporating johansen s algorithm for the maximum likelihood estimates and the likelihood ratio tests using the wald test statistics the paper also provides a method of confidence set construction for the causal measures the approach proposed in the paper includes testing granger s non causality as an instance of its multiple applications as an illustration the paper presents a characterization of the causal structure of the recent japanese macroeconomy on the basis of the proposed method and the derived evidence

canonical granger causality between regions of interest estimating and modeling functional connectivity in the brain is a challenging problem with potential applications in the understanding of brain organization and various neurological and neuropsychological conditions an important objective in connectivity analysis is to determine the connections between regions of interest in the brain however traditional functional connectivity analyses have frequently focused on modeling interactions between time series recordings at individual sensors voxels or vertices despite the fact that a single region of interest will often include multiple such recordings in this paper we present a novel measure of interaction between regions of interest rather than individual signals the proposed measure termed canonical granger causality combines ideas from canonical correlation and granger causality analysis to yield a measure that reflects directed causality between two regions of interest in particular canonical granger causality uses optimized linear combinations of signals from each region of interest to enable accurate causality measurements from substantially less data compared to alternative multivariate methods that have previously been proposed for this scenario the optimized linear combinations are obtained using a variation of a technique developed for optimization on the stiefel manifold we demonstrate the advantages of canonical granger causality in comparison to alternative causality measures for a range of different simulated datasets we also apply the proposed measure to local field potential data recorded in a macaque brain during a visuomotor task results demonstrate that canonical granger causality can be used to identify causal relationships between striate and prestriate cortexes in cases where standard granger causality is unable to identify statistically significant interactions

granger causality and information flow in multivariate processes the multivariate versus bivariate measures of granger causality were considered granger causality in the application to multivariate physiological time series has the meaning of the information flow between channels it was shown by means of simulations and by the example of experimental electroencephalogram signals that bivariate estimates of directionality in case of mutually interdependent channels give erroneous results therefore multivariate measures such as directed transfer function should be used for determination of the information flow

does sunspot numbers cause global temperatures a reconsideration using non parametric causality tests abstract in a recent paper gupta et al 2015 analyzed whether sunspot numbers cause global temperatures based on monthly data covering the period 1880 1 2013 9 the authors find that standard time domain granger causality test fails to reject the null hypothesis that sunspot numbers do not cause global temperatures for both full and sub samples namely 1880 1 1936 2 1936 3 1986 11 and 1986 12 2013 9 identified based on tests of structural breaks however frequency domain causality test detects predictability for the full sample at short 2 2 6 months cycle lengths but not the sub samples but since full sample causality cannot be relied upon due to structural breaks gupta et al 2015 conclude that the evidence of causality running from sunspot numbers to global temperatures is weak and inconclusive given the importance of the issue of global warming our current paper aims to revisit this issue of whether sunspot numbers cause global temperatures using the same data set and sub samples used by gupta et al 2015 based on an nonparametric singular spectrum analysis ssa based causality test based on this test we however show that sunspot numbers have predictive ability for global temperatures for the three sub samples over and above the full sample thus generally speaking our non parametric ssa based causality test outperformed both time domain and frequency domain causality tests and highlighted that sunspot numbers have always been important in predicting global temperatures

comparing alternative tests of causality in temporal systems abstract this paper discusses eight alternative tests of the absence of casual ordering all of which are asymptotically valid under the null hypothesis in the sense that their limiting size is known their behavior under alternatives is compared analytically using the concept of approximate slope and these results are supported by the outcomes of monte carlo experiments the implications of these comparisons for applied work are unambiguous wald variants of a test attributed to granger and a lagged dependent variable version of sim s test introduced in this paper are equivalent in all relevant respects and are preferred to the other tests discussed

validity of time reversal for testing granger causality inferring causal interactions from observed data is a challenging problem especially in the presence of measurement noise to alleviate the problem of spurious causality haufe 2013 proposed to contrast measures of information flow obtained on the original data against the same measures obtained on time reversed data they show that this procedure time reversed granger causality trgc robustly rejects causal interpretations on mixtures of independent signals while promising results have been achieved in simulations it was so far unknown whether time reversal leads to valid measures of information flow in the presence of true interaction here we prove that for linear finite order autoregressive processes with unidirectional information flow between two variables the application of time reversal for testing granger causality indeed leads to correct estimates of information flow and its directionality using simulations we further show that trgc is able to infer correct directionality with similar statistical power as the net granger causality between two variables while being much more robust to the presence of measurement noise

estimation of the generalized prediction error variance of a multiple time series abstract for a multivariate stationary time series we propose a nonparametric estimator for its generalized prediction error variance using a multivariate analog of the szego kolmogorov formula replacing the integral by a sum and replacing the unknown spectral density matrix by a consistent estimator asymptotic normality of this estimator is established and its small sample behavior is assessed through simulation and application to two real data sets these examples show that the proposed method works reasonably well in comparison with parametric models in contrast to the univariate case where smoothing the periodogram is optional and generally not recommended in the multivariate case this smoothing is a necessity because the raw periodogram is a matrix of rank one to obtain a consistent estimator of a full rank spectral density matrix one must necessarily choose larger bandwidths for higher dimensional time series a bias correction factor is computed using the asymptotic properties of our propo

joint analysis of spikes and local field potentials using copula recent technological advances which allow for simultaneous recording of spikes and local field potentials lfps at multiple sites in a given cortical area or across different areas have greatly increased our understanding of signal processing in brain circuits joint analysis of simultaneously collected spike and lfp signals is an important step to explicate how the brain orchestrates information processing in this contribution we present a novel statistical framework based on gaussian copula to jointly model spikes and lfp in our approach we use copula to link separate marginal regression models to construct a joint regression model in which the binary valued spike train data are modeled using generalized linear model glm and the continuous valued lfp data are modeled using linear regression model parameters can be efficiently estimated via maximum likelihood in particular we show that our model offers a means to statistically detect directional influence between spikes and lfp akin to granger causality measure and that we are able to assess its statistical significance by conducting a wald test through extensive simulations we also show that our method is able to reliably recover the true model used to generate the data to demonstrate the effectiveness of our approach in real setting we further apply the method to a mixed neural dataset consisting of spikes and lfp simultaneously recorded from the visual cortex of a monkey performing a contour detection task

hyponormality and subnormality of block toeplitz operators abstract in this paper we are concerned with hyponormality and subnormality of block toeplitz operators acting on the vector valued hardy space h c n 2 of the unit circle first we establish a tractable and explicit criterion on the hyponormality of block toeplitz operators having bounded type symbols via the triangularization theorem for compressions of the shift operator second we consider the gap between hyponormality and subnormality for block toeplitz operators this is closely related to halmos s problem 5 is every subnormal toeplitz operator either normal or analytic we show that if is a matrix valued rational function whose co analytic part has a coprime factorization then every hyponormal toeplitz operator t whose square is also hyponormal must be either normal or analytic third using the subnormal theory of block toeplitz operators we give an answer to the following toeplitz completion problem find the unspecified toeplitz entries of the partial block toeplitz matrix a u u so that a becomes subnormal where u is the unilateral shift on h 2

on complex symmetric toeplitz operators abstract in this paper we give a characterization of a complex symmetric toeplitz operator t on the hardy space h 2 moreover if t is a complex symmetric toeplitz operator we provide a necessary and sufficient condition for t to be normal finally we investigate these t with finite symbols

a comparison of least squares and conditional maximum likelihood estimators under volume endpoint censoring in tumor growth experiments measurements in tumor growth experiments are stopped once the tumor volume exceeds a preset threshold a mechanism we term volume endpoint censoring we argue that this type of censoring is informative further least squares ls parameter estimates are shown to suffer a bias in a general parametric model for tumor growth with an independent and identically distributed measurement error both theoretically and in simulation experiments in a linear growth model the magnitude of bias in the ls growth rate estimate increases with the growth rate and the standard deviation of measurement error we propose a conditional maximum likelihood estimation procedure which is shown both theoretically and in simulation experiments to yield approximately unbiased parameter estimates in linear and quadratic growth models both ls and maximum likelihood estimators have similar variance characteristics in simulation studies these properties appear to extend to the case of moderately dependent measurement error the methodology is illustrated by application to a tumor growth study for an ovarian cancer cell line

block toeplitz operators with rational symbols ii in this paper we derive a formula for the rank of the self commutator of hyponormal block toeplitz operators t with matrix valued rational symbols in via the classical hermite fej r interpolation problem

a general statistical analysis for fmri data we propose a method for the statistical analysis of fmri data that seeks a compromise between efficiency generality validity simplicity and execution speed the main differences between this analysis and previous ones are a simple bias reduction and regularization for voxel wise autoregressive model parameters the combination of effects and their estimated standard deviations across different runs sessions subjects via a hierarchical random effects analysis using the em algorithm overcoming the problem of a small number of runs session subjects using a regularized variance ratio to increase the degrees of freedom

testing causality between two vectors in multivariate autoregressive moving average models abstract in the analysis of economic time series a question often raised is whether a vector of variables causes another one in the sense of granger most of the literature on this topic is concerned with bivariate relationships or uses finite order autoregressive specifications the purpose of this article is to develop a causality analysis in the sense of granger for general vector autoregressive moving average arma models we give a definition of granger noncausality between vectors which is a natural and simple extension of the notion of granger noncausality between two variables in our context this definition is shown to be equivalent to a more complex definition proposed by tjostheim for the class of linear invertible processes we derive a necessary and sufficient condition for noncausality between two vectors of variables when the latter do not necessarily include all the variables considered in the analysis this result is then specialized to the class of stationary invertible arma process

a jackknife approach to quantifying single trial correlation between covariance based metrics undefined on a single trial basis the quantification of covariance between neuronal activities functional connectivity requires the observation of correlated changes and therefore multiple observations the strength of such neuronal correlations may itself undergo moment by moment fluctuations which might e g lead to fluctuations in single trial metrics such as reaction time rt or may co fluctuate with the correlation between activity in other brain areas yet quantifying the relation between moment by moment co fluctuations in neuronal correlations is precluded by the fact that neuronal correlations are not defined per single observation the proposed solution quantifies this relation by first calculating neuronal correlations for all leave one out subsamples i e the jackknife replications of all observations and then correlating these values because the correlation is calculated between jackknife replications we address this approach as jackknife correlation jc first we demonstrate the equivalence of jc to conventional correlation for simulated paired data that are defined per observation and therefore allow the calculation of conventional correlation while the jc recovers the conventional correlation precisely alternative approaches like sorting and binning result in detrimental effects of the analysis parameters we then explore the case of relating two spectral correlation metrics like coherence that require multiple observation epochs where the only viable alternative analysis approaches are based on some form of epoch subdivision which results in reduced spectral resolution and poor spectral estimators we show that jc outperforms these approaches particularly for short epoch lengths without sacrificing any spectral resolution finally we note that the jc can be applied to relate fluctuations in any smooth metric that is not defined on single observations

granger causality for state space models granger causality has long been a prominent method for inferring causal interactions between stochastic variables for a broad range of complex physical systems however it has been recognized that a moving average ma component in the data presents a serious confound to granger causal analysis as routinely performed via autoregressive ar modeling we solve this problem by demonstrating that granger causality may be calculated simply and efficiently from the parameters of a state space ss model since ss models are equivalent to autoregressive moving average models granger causality estimated in this fashion is not degraded by the presence of a ma component this is of particular significance when the data has been filtered downsampled observed with noise or is a subprocess of a higher dimensional process since all of these operations commonplace in application domains as diverse as climate science econometrics and the neurosciences induce a ma component we show how granger causality conditional and unconditional in both time and frequency domains may be calculated directly from ss model parameters via solution of a discrete algebraic riccati equation numerical simulations demonstrate that granger causality estimators thus derived have greater statistical power and smaller bias than ar estimators we also discuss how the ss approach facilitates relaxation of the assumptions of linearity stationarity and homoscedasticity underlying current ar methods thus opening up potentially significant new areas of research in granger causal analysis

model selection criteria for image restoration in this brief the image restoration problem is approached as a learning system problem in which a model is to be selected and parameters are estimated although the parameters which correspond to the restored image can easily be obtained their quality depend heavily on a proper choice of the regularization parameter that controls the tradeoff between fidelity to the blurred noisy observed image and the smoothness of the restored image by analogy between the model selection philosophy that constitutes a fundamental task in systems learning and the choice of the regularization parameter two criteria are proposed in this brief for selecting the regularization parameter these criteria are based on bayesian arguments and the kullback leibler divergence and they can be considered as extensions of the bayesian information criterion bic and the akaike information criterion aic for the image restoration problem

the decomposition and measurement of the interdependency between second order stationary processes for a given pair of multivariate stationary processes the process of one way effect is extracted from each of the processes each process is decomposed into two orthogonal processes namely into the process generated by the one way effect of the other process and the process orthogonal to it based on the decomposition three measures characterizing the interdependency of the pair of processes are introduced they are the measure of association the measure of one way effect and the measure of reciprocity each of the measures is defined as overall as well as frequencywise measure the paper shows that the measure of association is equal to the sum of the others it discusses the relationships of those measures to the ones proposed by gel fand yaglom and by geweke