1. Latent Causal Modelling of Neuroimaging Data Informatics and Mathematical Modeling Morten Mørup 1 1 Cognitive Systems, DTU Informatics, Denmark, 2 Danish Research Centre for Magnetic Resonance, email: {mm,khm,lkh}@imm.dtu.dk Lars Kai Hansen 1 Kristoffer Hougaard Madsen 2 x4x4 s1s1 s2s2 Latent Source Measurement Channel Benefits of SLCM over DTF:  SLCM can potentially perform dimensionality reduction resulting in fewer latent sources than observed measurement voxels/channels.  Constraints on the causal relations can be directly imposed on A(t) such as sparsity [2,7] and restricting the transfer function to specific delays.  Spatial regions that are caused by the d th source s d (t) are automatically grouped in a d (t).  SLCM can handle instantaneous mixing whereas DTF is hard to interpret in case of instantaneous propagation between voxels/channels [4,5].  SLCM can naturally handle overcomplete representations, i.e. I>>T.  The estimation of SLCM is a non-convex problem Synthetic 64 channel EEG dataset. Left panel: The simulated A(t) contains two latent sources (component 1 and 2) and two components granger causing other channels (component 3 and 4). Middle panel: Estimated A(t) from a SLCM analysis. Right panel: The corresponding Granger analysis based on the DTF approach [5]. From the power of the estimated input functions e(t) of each channel it can be seen that 9 of the 64 input functions are active. The most active input functions are the input functions of channel 11 and 1 corresponding to the two channels of component 3 and 4 that were generated to Granger cause the remaining channels. Below are given the significant maximal autocorrelations (on an a=1% level) between the input functions e(t) and true simulated latent sources s(t). The estimated inputs functions to channel 11 and 1 are (correctly) significantly correlated to the latent source 3 and 4 respectively. As the DTF analysis is unable to correctly account for the dynamics of component 1 and 2 due to the instantaneous propagation between channels the information of these underlying two latent sources are arbitrarily distributed to several of the channels that observe these sources. Real 64 channel EEG dataset based on visual paradigm. From the SLCM analysis a four component model was extracted. The space-time dynamics of the most prominent first component pertain to visual activation and indicate a flow from left to right occipital areas and later to more central regions. Channel Specific Input Functions Noise Latent Sources e2e2 x2x2 e1e1 e3e3 e4e4 e5e5 e6e6 x3x3 x1x1 x6x6 x5x5 Channel Specific Input Function SLCM Sparse Latent Causal Modelling (SLCM) We will consider latent input functions s(t) based on the following convolutional representation Where e i (t) is residual noise. Thus measurements are caused by underlying latent sources rather than causes constrained to be the measurement channels themselves. We note that this representation corresponds to the convolutive ICA model [3, 8]. Granger Causality and the Directed Transfer Function (DTF) We are interested in causality based on the common sense notion that causes always precede their effects. Causal inference in neuroimaging data is typically approached by Granger causality [4] and the related directed transfer function (DTF) method [5]., e.g. by fitting the following multivariate autoregressive model to the space-time data x(t) [5] In the frequency domain these systems of equations correspond to [5] Where H(f)=(I-Q(f))-1 denotes the transfer function. Ie. h i,j (f) denotes the influence of j th on i th channel/voxel at frequency f. In the time domain this can equivalently be written as where e j (t) is the input signal to channel j and h i,j (t) the influence of channel j to channel i at delay t. Inference for SLCM We use Bayesian inference and impose sparseness priors on the filter coefficients in the form of automatic relevance determination (ARD) [1,7], this includes inference for the model order D. Transfer Functions