(A) This example presents a schematic representation of diffusion map embedding. It illustrates the calculation of the principal functional gradient of 4 cerebellar voxels (red, green, blue, magenta) based on their functional connectivity with two target cerebellar voxels (yellow, orange). (B) Connectivity from each cerebellar voxel (red, green, blue, magenta) to the two target cerebellar voxels (yellow, orange) is represented as a two-dimensional vector. (C) All vectors can be represented in the same two-dimensional space. (D) Cosine distance between each pair of vectors is calculated, and (E) an affinity matrix is constructed as (1-cosine distance) for each pair of vectors. This affinity matrix represents the similarity of the connectivity patterns of each pair of voxels. (F) A Markov chain is constructed using information from the affinity matrix. Information from the affinity matrix is thus used to represent the probability of transition between each pair of vectors. In this way, there will be higher transition probability between pairs of voxels with similar connectivity patterns. This probability of transition between each pair of vectors can be analyzed as a symmetric transformation matrix, thus allowing the calculation of eigenvectors. (G) Eigenvectors derived from this transformation matrix represent the principal orthogonal directions of transition between all pairs of voxels. Here we illustrate the first resulting component of this analysis – the principal functional gradient of our four cerebellar voxels (red, green, blue, magenta) based on their connectivity with our two target cerebellar voxels (yellow, orange) progresses from the blue, to the green, to the magenta, to the red voxel. (H) This order is mapped back into our cerebellum map, allowing us to generate functional neuroanatomical descriptions. Of note, our cerebellar functional gradients were calculated using functional connectivity values of each cerebellar voxel with the rest of cerebellar voxels (rather than between four voxels and only two target cerebellar voxels, as in this example). Vectors in our analysis thus possessed many more than just two dimensions, but cosine distance can also be calculated between pairs of high-dimensional vectors. Diffusion map embedding using functional connectivity values from each cerebellar voxel to all cerebellar voxels thus captures the principal gradients of cerebellar functional neuroanatomy.