  We propose new small-sphere distributional families for modeling multivariate
directional data on $(\mathbb{S}^{p-1})^K$ for $p \ge 3$ and $K \ge 1$. In a
special case of univariate directions in $\Re^3$, the new densities model
random directions on $\mathbb{S}^2$ with a tendency to vary along a small
circle on the sphere, and with a unique mode on the small circle. The proposed
multivariate densities enable us to model association among multivariate
directions, and are useful in medical imaging, where multivariate directions
are used to represent shape and shape changes of 3-dimensional objects. When
the underlying objects are rotationally deformed under noise, for instance,
twisted and/or bend, corresponding directions tend to follow the proposed
small-sphere distributions. The proposed models have several advantages over
other methods analyzing small-circle-concentrated data, including inference
procedures on the association and small-circle fitting. We demonstrate the use
of the proposed multivariate small-sphere distributions in analyses of
skeletally-represented object shapes and human knee gait data.
