## AbstractTests to determine the dependence or independence of random variables are well established in statistical analysis: the mutual information (MI) is one example, while another approach (used in some methods for independent component analysis) is to assume a parametric model of the underlying probability density. We propose a novel non-parametric independence criterion, the constrained covariance (COCO), which is a measure of the covariance between (infinite dimensional) vectors of nonlinear functions of the observations. When these vectors are members of universal reproducing kernel Hilbert spaces, we demonstrate that the constrained covariance is zero only when the random variables being tested are independent (as with the mutual information). In practice, we do not need to deal explicitly with these infinite dimensional mappings, but only with the inner products between the mapped points (given by a kernel function). Thus, COCO can be efficiently computed on the basis of a finite sample. We further show that COCO behaves in a reasonable manner in the case of dependent random variables, in that the dependency detected becomes small as the density generating the random variables becomes less smooth. This is beneficial in that if we get a small sample from a density in which the dependence is encoded in a highly non-smooth way, then we cannot empirically distinguish this dependence level from independence (zero). With a larger sample size, the empirical estimate of COCO will have a smaller variance, and small (but non-zero) COCO values will then differ significantly from zero. Moreover, the rate at which the population COCO drops as the density becomes non-smooth can be controlled directly through the choice of the non-linear mapping (which is itself determined by the kernel). Our main focus is a neuroscience application of dependence detection with COCO. A number of groups have begun examining the interactions between neural systems using fMRI in humans. Compared with this work, the recent study of BOLD fMRI in the macaque monkey using high field (4.7T \\& 7T) scanners has resulted in substantial increases in spatial and temporal resolution, when measuring brain activity patterns resulting from various stimuli. We apply COCO to these high resolution data so as to detect dependence between BOLD responses within the visual cortex, comparing with both the mutual information and with the cross correlation (the latter not being a dependence test in the general sense, but measuring only second order effects). The variation in dependence between voxels was studied with all three methods, as a function of average distance between voxels (in other words, we grouped together all pairs of voxels an equal distance from each other; we then clustered these pairs so as to draw together voxel pairs with similar distances). To compare the different dependence estimation techniques, we subtracted a baseline dependence from each of the dependence measures, which was obtained by averaging the dependence between the V1 region and a test region of the brain, the latter being unrelated to visual processing. The dependence amplitudes were then divided by the standard deviation in the average dependence between V1 and this test region. The observed fMRI signals were contaminated with a breathing component. Since the macaque monkey was under general anaesthetic during data acquisition, breathing was mechanically assisted, and had a constant frequency of approximately 0.4Hz. We modelled this breathing as being of constant amplitude and linearly superposed on the haemodynamic response. This model is motivated by the narrowness of the spectral peaks at the breathing frequency and harmonics, which suggests that any amplitude modulation of the breathing signal is of very low frequency, and can be assumed effectively constant. Thus, while we could not directly recover the true breathing contamination at each voxel, we were able to use the decrease in the spectral peak at the breathing frequency, averaged across all voxels, as a measure of success in removing the breathing artefacts. Only voxels near large blood vessels were contaminated by the breathing signal, and thus a threshold test was applied to the spectrum at each voxel, to test whether a substantial breathing component was visible. The breathing was removed by projection in the time domain, as band-pass filtering would have eliminated a greater portion of the spectral components due to the haemodynamic activity. Comparing the dependence measures before and after breathing removal shows significant effects of respiratory artefacts on the high order dependence vs. distance curves (COCO and MI): this finding suggests extreme caution for studies in humans, in which respiration-induced signals cannot easily be modelled due to low temporal sampling rates, as well as variable respiration frequency and amplitude. Prior to breathing removal, COCO and the MI overestimate the dependence between voxels (the breathing artefacts being a source of considerable similarity), as does the correlation, though to a lesser extent. This can be explained by phase shifts between the breathing contamination observed at different voxels, which reduce the correlation but have less effect on more general measures of dependence. The high order dependence curves also flatten out after about 5mm once breathing is removed, but continue to decay with distance when breathing is present. By contrast, the correlation prior to breathing removal is constant (to within observational uncertainty) after about 2mm; following breathing removal, however, the point at which it flattens out is more difficult to determine. Finally, compared with the MI, COCO at short distances is a larger multiple of the standard deviation in test region dependence, which might make COCO a more reliable measure of such short range dependencies. On the other hand, both the MI and the correlation fall to a baseline level of activity greater than that in the test region, which COCO does not detect.
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