电子科技大学 生命科学与技术学院非线性多变量处理方法非线性多变量处理方法汇报内容n线性和非线性信号分析方法与应用n近期工作研究进展n工作计划一、信号分析方法：nCross-correlation functionnCoherencenNonlinear correlation coefficientnGranger causalitynPartial coherencenPartial directed coherencenPhase synchronizationnComparing the different approaches1、Cross-correlation functionnDefinitionn n-1-1 1. 1.n nEstimatesEstimates：MatlabMatlab function: c= function: c=xcorr(x,yxcorr(x,y) )Applications to neurophysiologynThe first approaches to correlation measurements between two simultaneously measured EEG signals were made more than fifty years ago (Brazier and Barlow, 1956;Brazier and Casby, 1952).nHowever, the cross-correlation function and its variant, the cross-correlogram histogram (Perkel et al., 1967) remains one of the mostly used measures to reveal the temporal coherence in the firing of cortical neurons from their spike trains (Brody, 1999; Nowak and Bullier, 2000).2、CoherencenDefinition 0 1. linearly independent; 0 1. linearly independent; maximum linear correlation.maximum linear correlation. EstimatesEstimates：MatlabMatlab function: function: n nCross Spectral Density estimate: Cross Spectral Density estimate: csdcsd()()n nPower Spectral Density estimate: Power Spectral Density estimate: psdpsd()()n na parametric approach: a parametric approach: pyulear(),pburgpyulear(),pburg()()FactorsnThe length of the data segment for analysis, which must be short enough to satisfy the condition of stationarity, and long enough to provide good frequency resolution.nReference electrodes and volume conduction.Applications to neurophysiologynCoherence was first applied to EEG signals more than forty years ago (Adey et al., 1967a; Brazier, 1968; Walter and Adey, 1963; Walter et al., 1966).nwe mention a few key articles that reviewed the applications of coherence to neural data (Dumermuth and Molinari, 1991; French and Beaumont, 1984; Shaw, 1984; Thatcher et al., 1986; Zaveri et al., 1999).3、Nonlinear correlation coefficientnDefinition: which describes the dependency of X on Y in a most general way without any direct specification of the type of relationship between them.nwhere f is the linear piecewise approximation of the nonlinear regression curve.n把所有的x分成多段（M），求平均，对应的y 也求段内平均，把这M个点连接起来就构成了 函数关系f。n同样可以计算出Asymmetry, time delay and direction in couplingn0 (Y is completely independent of X) to 1 (Y is fully determined by X).nthe relationship between these signals is linear, and this measure approximates the squared linear regression coefficient .indicates the degree of asymmetry of the nonlinear coupling.nthe delay my|x at which the maximum value is used as an estimate of the time delay between the signals. If X causes Y, my|x will be positive, so that the difference m =my|x mx|y will be also positive. A robust measure on the direction of coupling:1:xy,-1:yx,0:xy.ApplicationsnIts applications have been confined exclusively to epileptic EEG data analysis (Meeren et al., 2002; Pijn et al., 1990; Wendling et al., 2001).4、Granger causalitynA question of great interest is whether there exists a causal relation between two brain regions without any specific information on direction.nOne of the first attempts involved the method of structural equation modeling (Asher, 1983).Definitionnfor two simultaneously measured signals, if one can predict the first signal better by incorporating the past information from the second signal than using only information from the first one, then the second signal can be called causal to the first one (Wiener, 1956).n(Granger, 1969) Granger argued that if X is influencing Y, then adding past values of the first variable to the regression of the second one will improve its prediction performance, which can be assessed by comparing the univariate and bivariate fitting of the AR models to the signals. DefinitionnThus, for the univariate case, one has:On the other hand, for bivariate AR modeling,nIf VX|XYVX|X then Y causes X in the sense of Granger causality. The Granger causality of Y to X can be quantified as:ApplicationsnAlthough Granger causality was introduced more than thirty five years back, most of its applications to neural data analysis are within the last six years. (Freiwald et al., 1999).ninvestigate the role of bottom-up and top- down interactions in a go/no-go task (Bernasconi et al., 2000) or in a stimulus expectancy task (Salazar et al., 2004).5、Partial coherencenThe first extension of bivariate analysis was made by incorporating a third signal into the estimation of a new coherence measure, termed as partial coherence.n For signals X, Y, and Z, the underlying point is to subtract linear influences from other processes to obtain the partial cross-spectrum between X and Y given all the linear information of Z :5、Partial coherencenpartial cross-spectrum between X and Y given all the linear information of Z :nif Z contributes to the linear interdependence betwe