Multivariate fmri Analysis using Canonical Correlation Analysis instead of Classifiers, Comment on Todd et al.

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1 Multivariate fmri Analysis using Canonical Correlation Analysis instead of Classifiers, Comment on Todd et al. Anders Eklund a, Hans Knutsson bc a Virginia Tech Carilion Research Institute, Virginia Tech, Roanoke, USA b Division of Medical Informatics, Department of Biomedical Engineering, Linköping University, Linköping, Sweden c Center for Medical Image Science and Visualization (CMIV), Linköping University, Linköping, Sweden Abstract Multivariate pattern analysis (MVPA) is a popular method for making inference about functional magnetic resonance imaging (fmri) data. One approach is to train a classifier with voxels within a certain radius from the center voxel, to classify between different brain states. This approach is commonly known as the searchlight algorithm. As recently pointed out by Todd and colleagues, inference at the group level can however be confounded by the fact that the direction of the effect is lost if the per subject classification performance is used to generate group results. Here we show that canonical correlation analysis (CCA) can in some aspects be a better approach to multivariate fmri analysis, than classification based analysis (CBA). Keywords: fmri, GLM, CCA, MVPA 1. Introduction Todd et al. (2013) describe a problem that exists for classification based multivariate pattern analysis (MVPA) of functional magnetic resonance imaging (fmri) data at the group level. The paper starts by stating that fmri analysis can be characterized in terms of two broad approaches; single-voxel general linear model based analysis (GLMA) (Friston et al., 1995b) and classification based analysis using MVPA (specifically searchlight MVPA (Kriegeskorte et al., 2006)). The MVPA approach can be more powerful, since information adaptively combined from several voxels can yield a higher sensitivity than univariate approaches. This is of uttermost importance as many fmri studies are severely underpowered (Button et al., 2013). As stated by Todd et al. (2013), a drawback of classification based MVPA is however that the sign, or direction, of the underlying effect being tested is lost if the classification performance is carried on to the group level analysis, which can lead to confounding results. There seems to be a common belief in the neuroimaging field that a classifier has to be used for multivariate fmri analysis. Here we therefore show that the problem mentioned by Todd et al. (2013) can be solved by instead using canonical correlation analysis (CCA), and that this approach has several advantages compared to classification based MVPA. Since fmri analysis based on both CCA and classifiers use information Corresponding author Virginia Tech Carilion Research Institute 2 Riverside Circle Roanoke Virginia, USA address: andek034@gmail.com (Anders Eklund a ) from several voxels in an adaptive way, they can both be considered as MVPA approaches. We will therefore refer to classification based MVPA as classification based analysis (CBA). 2. GLMA vs CCA vs CBA While most fmri researchers use GLMA or CBA, we prefer to use an approach that combines the good aspects of these two methods. This approach is, however, less known in the fmri field. It is for example not mentioned in a recent review about early multivariate approaches in fmri (Haxby, 2012). Canonical correlation analysis (CCA) (Hotelling, 1936) finds the best linear combination of two multivariate variables, while the GLM used for fmri is applied to one univariate and one multivariate variable. In fmri, CCA can for example be used to find the best linear combination of temporal and spatial regressors simultaneously (Friman et al., 2001, 2003; Nandy and Cordes, 2003; Friman, 2004; Nandy and Cordes, 2004; Rydell et al., 2006; Ragnehed et al., 2009; Eklund et al., 2011; Cordes et al., 2012; Jin et al., 2012; Eklund et al., 2012b). The temporal regressors for CCA can be the same as for GLMA and the spatial regressors can for example be the time series of neighbouring voxels. CCA-based fmri analysis can for example be performed with a plugin for SPM 1, the authors are also working on an fmri software package with GPU acceleration. An illustrative comparison between GLMA, CCA and CBA is given in Figure 1. CCA is the only approach that provides true spatiotemporal analysis. Canonical variates analysis (CVA) is closely related to CCA and was also early applied to fmri (Friston et al., 1995a; 1 Preprint submitted to figshare July 29, 2013

2 Fletcher et al., 1996; Poline et al., 1997; Strother et al., 1996; LaConte et al., 2005). The main difference between CCA and CVA is that the covariance matrix for the spatial regressors is in CVA subjected to a principal component analysis (PCA); to guarantee that the matrix has full rank. This can however lead to the removal of important information. Another option to estimate the covariance matrix is to use partial least squares (PLS), as proposed by McIntosh et al. (1996). GLMA is for fmri a temporal method that finds the best combination of temporal regressors for each voxel independently. fmri analysis based on GLMA or CCA can rather easily adjust for the unknown BOLD delay in each voxel, for example by including the temporal derivative of the stimulus paradigm as a second temporal regressor. CBA can, however, only handle a single discrete temporal regressor, representing the class of each time point (e.g. activity or rest), as CBA is for fmri mainly a spatial method. This means that the CBA approach has to use a preset BOLD delay, being an integer number of time samples. Spatio-temporal CBA is multivariate in both space and time, i.e. similar to CCA based fmri analysis, and was recently proposed by Fogelson et al. (2011) and Rao et al. (2011). CCA based fmri analysis can, just as CBA, yield a higher statistical power than GLMA by adaptively combining several voxels. In contrast to CBA, CCA is a multivariate approach that also has the nice properties of GLMA. The direction of the underlying effect, e.g. the sign of the canonical correlation or the sign of a specific temporal or spatial weight, can for CCA be carried on from the first level analysis to the second level analysis. Since CCA is a rather old approach that originates from statistics, while CBA is a new method emerging from machine learning, there also exists a much larger number of applications and extensions of CCA. For example, Gaussian random field theory is available for CCA to correct for multiple comparisons in fmri (Worsley et al., 2004) and work has been conducted on restricted CCA (Das and Sen, 1994), for example useful if one wishes to apply restrictions to the possible weights (Friman et al., 2003; Cordes et al., 2012). Extensions of CCA can also work with more than two sets of variables (Kettenring, 1971; Gomez et al., 2004), making it possible to include other sources of information (e.g. electroencephalography (EEG) or diffusion tensor imaging (DTI) data) in the fmri analysis. See for example the work by Correa et al. (2010). CCA based fmri analysis has a rather low computational complexity, while CBA for each voxel often requires several iterations to find the best classifier weights. The main problem with CBA is however not the computational complexity of the classifier itself, but that some cross validation scheme is required to obtain a robust estimate of the classification performance. The main reason for this is the unrestricted use of a large number of voxels (e.g ) in a searchlight sphere. For leave-one-out cross validation, it is necessary to go through the data N times if there are N volumes. In contrast, CCA based fmri analysis using a smaller number of voxels, or filters (Friman et al., 2003; Friman, 2004), does not need any cross validation (neither does GLMA), and can thus potentially be N times faster than CBA for N volumes. CBA using filters, in- 2 stead of voxels directly, can potentially also become faster by lowering the degrees of freedom and thereby circumvent the need of cross validation. Both CCA and CBA can have a complicated null distribution, but this can today be solved by combining permutation tests (Nichols and Holmes, 2001) and graphics processing units (GPUs) (Eklund et al., 2012b, 2013). In our opinion, nonparametric approaches should be used as often as possible as they rely on a smaller number of assumptions. Non-parametric approaches have for fmri been shown to yield more valid results than commonly used parametric ones (Eklund et al., 2012a). The main differences and similarities between GLMA, CCA and CBA for fmri have been summarized in Table 1. CCA based fmri analysis clearly combines the advantages of GLMA and CBA. 3. Simulation To show that CCA indeed solves the problem described by Todd et al. (2013), a small simulation was performed. An fmri dataset with 3 x 3 x 3 voxels and 200 time points was created with activity from a simple on-off block design. The activity was inserted in the corners and the center voxel, such that a total of 9 voxels contained activity with different noise. The simulation was done for four artificial subjects, two having a positive response (compared to the design) and two having a negative response. GLMA, CCA and CBA (using the the builtin support vector machine classifier in Matlab) were then used to calculate a measure of brain activity for the center voxel for the four subjects, the code used for the simulation is available online 2. The simulation was repeated for 100 runs and the mean weight patterns were calculated for CCA and CBA. The resulting brain activity measurements for GLM, CCA and CBA are presented in Figures 2 and 3. The mean weight patterns for CCA and CBA are given in Figure 4. From this small simulation the following observations were made Both GLMA and CCA can provide sign information to separate the two subject groups with a positive and negative response, which CBA is unable to. The CCA approach finds a good approximation of the inserted spatial pattern with a single run, while the CBA approach needs to average the weights over several runs to obtain a good estimate. CCA is with our code approximately times faster than CBA, and this does not even include cross-validation for CBA. This large speedup can, however, be due to the fact that the built-in Matlab implementation of the support vector machine classifier is sub-optimal. If leave-oneout cross validation is used with CBA for 100 samples, the CCA approach would be about 10,000-20,000 times faster. 2

3 Figure 1: An illustrative comparison between GLMA, CCA and CBA for fmri analysis. GLMA combines a number of temporal regressors for each voxel time series independently (here the paradigm convolved with the hemodynamic response function and its temporal derivative), the resulting combination is shown in red. CCA combines spatial and temporal regressors at the same time. The temporal regressors are in this case the same as for the GLM and the spatial regressors are here simply time series from neighbouring voxels. The resulting temporal combination is given in red and the resulting spatial combination is given in green. CBA creates a classifier in each voxel by also combining neighbouring voxel time series, but instead only uses one temporal regressor (the class of each time point). 3 The resulting spatial combination is given in green. CCA is the only approach that combines both spatial and temporal regressors and is thereby a good option for spatio-temporal fmri analysis. Due to space limitations, only three neighbouring time series are here shown for CCA and CBA, but the green time series are actually a linear combination of time series from the center voxel and its 26 neighbours.

4 Table 1: The main differences and similarities between general linear model analysis (GLMA), canonical correlation analysis (CCA) and classification based analysis (CBA) for fmri. CBA has a higher computational complexity since some kind of cross-validation scheme normally is applied. Property / Method GLM CCA CBA Sign information for direction of effect Yes Yes No Number of temporal regressors Arbitrary Arbitrary 1 Number of spatial regressors 1 Arbitrary Arbitrary Adaptive to BOLD delay Yes Yes No Gaussian random field theory exists Yes Yes No Computational complexity Low Low Medium - High Null distribution Simple Complicated Complicated The processing times for GLMA and CCA are constant regardless of the noise level, while training of the SVM classifier takes a longer time when more noise is present. 4. Conclusion We have made a small comparison between CCA and CBA, showing that CCA based fmri analysis can solve the problem of confounding group analysis that exists with CBA, while still being able to detect distributed patterns. References Button, K., Ioannidis, J., Mokrysz, C., Nosek, B., Flint, J., Robinson, E., Munafo, M., Power failure: why small sample size undermines the reliability of neuroscience. Nature Reviews Neuroscience 14, Cordes, D., Jin, M., Curran, T., Nandy, R., Optimizing the performance of local canonical correlation analysis in fmri using spatial constraints. Human brain mapping 33, Correa, N., Eichele, T., Adali, T., Li, Y.O., Calhoun, V., Multi-set canonical correlation analysis for the fusion of concurrent single trial ERP and functional MRI. NeuroImage 50, Das, S., Sen, P., Restricted canonical correlations. Linear Algebra and its Applications 210, Eklund, A., Andersson, M., Josephson, C., Johannesson, M., Knutsson, H., 2012a. 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5 Figure 2: A comparison of activation values for GLMA, CCA and CBA, for four subjects and 100 runs per subject. The first 200 values represent subjects 1 and 2 (which had a positive effect), the last 200 values represent subjects 3 and 4 (which had a negative effect). Both GLMA and CCA can provide sign information to separate the two groups, which CBA cannot. The mean activity over subjects is for GLMA and CCA close to zero, while it is clearly not zero for CBA. For visualization purposes, the canonical correlation values were multiplied with a factor 5. Figure 3: A comparison of activation values from GLMA, CCA and CBA in terms of distributions. Both GLMA and CCA yield two distinct groups of activity values (representing the two subject groups with a positive or negative effect), while CBA yields a single group. 5

6 Figure 4: A comparison of pattern detection for CBA (using support vector machines) and CCA. The true pattern is activity in the corners and the center voxel, for a simulated brain consisting of 3 x 3 x 3 voxels. Top: Weights for a single run. Bottom: Weights averaged over 10 runs. 6