Joints

For several of the data sets, the diffusion gradients are duplicated for opposing PE-directions. This offers a way to assess the performance of eddy and also to compare it to a commonly used existing method (eddy_correct in FSL) that uses FLIRT (Jenkinson and Smith, 2001 (link)), a 12-dof affine transformation and correlation ratio as a cost-function to register the diffusion weighted images to a b = 0 image. Two images acquired with the same diffusion gradient will have the same contrast and any difference between them should be due to differences in distortions and/or measurement error (noise). We therefore ran eddy separately on the data with the two different PE-directions and then calculated the sum-of-squared differences for paired diffusion weighted images.
Data acquired with different PE-directions also differ with respect to different susceptibility-induced distortions and if not corrected these would dominate any comparison between the images. We therefore used RGM and pairs of b = 0 (where there will be no eddy current-induced distortions) with different PE-directions to estimate the susceptibility-induced off-resonance field and applied that to the images using spline-interpolation and Jacobian modulation (see section Resampling the images). These “susceptibility-only corrected” pairs were the baseline against which the eddy and eddy_correct methods were compared. The eddy_correct method was modified to use spline interpolation and also to be able to incorporate the susceptibility-field from RGM so as to allow for a single resampling into a space corrected for susceptibility, eddy currents and subject movement in the same way that eddy does.
A series of tests was run on the FMRIB (data sets $\mathcal{A}$ and $\mathcal{B}$ ) and the early HCP data ( $\mathcal{C}$ ) to evaluate different settings for the options in eddy. As described above these tests were performed by running eddy separately on the A → P and the P → A (or L → R and R → L in the case of the HCP 3 T data ( $\mathcal{C}$ )) data and then compared pairwise to assess how well the correction worked. These settings were
Estimation of GP hyperparameters There are several different options for determining the hyperparameters for the Gaussian process that model the diffusion signal. These are maximum marginal likelihood (MML), leave-one-out cross validation (CV) and Geissers's surrogate predictive probability (GPP). For each method data was extracted from 1000 random brain voxels and used for the estimation. Note that this random voxel selection potentially introduces a run-to-run variability to the eddy results, but which can be turned off by specifying a seed at the command level.
Q-space smoothing The GP can be seen as a smoothing operation in Q-space. We tested different levels of increased smoothing by multiplying the error-variance estimates (hyperparameter of the GP) by values ranging from 1 (no additional smoothing) to 10.
Spatial smoothing Data and predictions were smoothed with a Gaussian filter with FWHM ranging from 0 to 5 mm. N.B. that the filtering is applied only during the estimation phase and not to the final resampled results.
EC model Different models for the EC-induced fields corresponding to first (four parameters), second (ten parameters) and third (20 parameters) order polynomials were tested. See Appendix A for a complete description of the different models.
Second level modeling The EC-parameters were fitted to a first or second order polynomial at the end of each iteration.
Joint modeling of multi-shell data When having multi-shell data one can either correct each shell independently or one can model (and correct) them all simultaneously. The latter option is potentially better because the Gaussian process is able to use data from one shell when making predictions about another shell (Andersson and Sotiropoulos, 2015 ). To test that, we corrected the HCP 3 T data ( $\mathcal{C}$ ) for each shell individually and also jointly for all four shells.

The fitting of MEME to an alignment of coding sequences proceeds in three stages:
First, the codon model with an alignment-wide is fitted to the data using parameter estimates under a GTR nucleotide model as initial values. Although in some cases nucleotide branch lengths may be a good approximation to codon branch lengths [23] (link), [24] (link), recent results indicate that in other instances, nucleotide models can significantly underestimate branch lengths and possibly bias downstream inference [25] . The resulting maximum likelihood estimates, and , for each branch , are used in the site-by-site analyses in the next two steps. Thus we are assuming that the relative branch length and mutational bias parameters are shared across sites and are well approximated by those estimated under a simpler codon model. However, the absolute branch lengths also depend on the site- and model-specific rate parameters below.
Second, at each site, we first fit the alternative random effects model of lineage-specific selective pressure with two categories of : and (unrestricted). The probability ( in equation 1) that branch is evolving with , is , and the complementary probability that it is evolving with is . By equation 1, the phylogenetic likelihood at a site, marginalized over all possible joint assignments of , is equivalent to computing the standard likelihood function with the following mixture transition matrix for each branch :
Consequently, the alternative substitution model includes four parameters for each site, inferred jointly from all branches of the tree: and . These form the fixed effects component of the model. Estimating separately for each site accounts for the site-to-site variability in synonymous substitution rates [26] (link).
Lastly, at every site, we fit the model from the previous step, but with : our null model. Using simulated data, we determined that an appropriate asymptotic test statistic for testing most worst-case null of of is a mixture of and (see Text S1). Mixture statistics of this form often arise in hypothesis testing where model parameters take values on the boundaries of the parameter space, and closed-form expressions for mixing coefficients are difficult to obtain [27] .
Throughout the manuscript, we compare MEME to the fixed effects likelihood approach, introduced in [24] (link) (see Text S1 for motivation). The procedure used by FEL differs from MEME in that a single pair of rates are fitted at each site (no variation over branches) in Step 2, and the test in Step 3 is to determine if . Positive selection is inferred by FEL when and the p-value derived from the LRT is significant, based on the asymptotic distribution.