Robust Comparison of Kernel Densities on Spherical Domains

Abstract

While spherical data arises in many contexts, including in directional statistics, the current tools for density estimation and population comparison on spheres are quite limited. Popular approaches for comparing populations (on Euclidean domains) mostly involve a two-step procedure: (1) estimate probability density functions (pdf s) from their respective samples, most commonly using the kernel density estimator, and (2) compare pdf s using a metric such as the $\mathbb {L}^{2}$ norm. However, both the estimated pdf s and their differences depend heavily on the chosen kernels, bandwidths, and sample sizes. Here we develop a framework for comparing spherical populations that is robust to these choices. Essentially, we characterize pdf s on spherical domains by quantifying their smoothness. Our framework uses a spectral representation, with densities represented by their coefficients with respect to the eigenfunctions of the Laplacian operator on a sphere. The change in smoothness, akin to using different kernel bandwidths, is controlled by exponential decays in coefficient values. Then we derive a proper distance for comparing pdf coefficients while equalizing smoothness levels, negating influences of sample size and bandwidth. This signifies a fair and meaningful comparisons of populations, despite vastly different sample sizes, and leads to a robust and improved performance. We demonstrate this framework using examples of variables on $\mathbb {S}^{1}$ and $\mathbb {S}^{2}$, and evaluate its performance using a number of simulations and real data experiments.

Appendix

1.1 Proof that $S_{\kappa }$ is an Orthogonal Section

An orthogonal section $S_{\kappa }$ is a subset of $\mathbb {R}^{N + 1}$ (coefficient representation of densities) under the action of the group $\mathbb {R}$ (defined in Eq. 2.3 in the main paper) if: (i) one and only one element of every orbit $[\boldsymbol {c}]$ in $\mathbb {R}^{N + 1}$ presents in $S_{\kappa }$, and (ii) the set $S_{\kappa }$ is perpendicular to every orbit at the point of intersection. The last property means that if $S_{\kappa }$ intersects an orbit $[\boldsymbol {c}]$ at $\tilde {\boldsymbol {c}}$, then $T_{\tilde {\boldsymbol {c}}}(S_{\kappa }) \perp T_{\tilde {\boldsymbol {c}}}([\boldsymbol {c}])$. We need to verify the two properties: (1) The function $t \mapsto {\sum }_{n} e^{-2\lambda _{n} t} \lambda _{n} {c_{n}^{2}}$ is a strictly monotonically-decreasing function that ranges ($+\infty$, 0). Thus, for any $\boldsymbol {c} \in \mathbb {R}^{N + 1}$ and $\kappa > 0$, there exists a unique $t^{*}$ such that ${\sum }_{n} e^{-2\lambda _{n} t^{*}} \lambda _{n} {c_{n}^{2}} = \kappa$. (2) At any point c ∈ S_κ, the space normal to $S_{\kappa }$ (inside $\mathbb {R}^{N}$, notice that $\lambda _{0} = 0$) is a one-dimensional space spanned by the vector $\textbf {n}_{\boldsymbol {c}} = \{ \lambda _{1} c_{1}, \lambda _{2} c_{2}, \dots , \lambda _{N} c_{N}\}$. Let $\textbf {u}_{\boldsymbol {c}}$ denote the unit vector in the normal direction u_c = n_c/∥n_c∥. Since $S_{\kappa }$ is a level set of G, it is automatically perpendicular to $\textbf {u}_{\boldsymbol {c}}$ and $T_{c}([\boldsymbol {c}])$. In other words, the orbits are just the flow lines for the gradient vector field of the function G and since the level sets of a functional are perpendicular to the flow lines of gradient of that function, it follows that the S_κ is perpendicular to these orbits.

1.2 Path Straightening Algorithm on $S_{\kappa }$

Here we present the path straightening algorithm for calculating distances on $S_{\kappa }$. We first list the following basic tools for the path straightening algorithm.

• 1. Projection onto Mainfold $S_{\kappa }$: For any arbitrary point $\boldsymbol {c} \in \mathbb {R}^{N}$, we need a tool to project $\boldsymbol {c}$ to the nearest point in $S_{\kappa }$. One can find this nearest point by iteratively updating $\boldsymbol {c}$ according to $\boldsymbol {c} \mapsto \boldsymbol {c} + (\kappa - G(\boldsymbol {c}))\textbf {u}_{\boldsymbol {c}}$, until $G(\boldsymbol {c}) = \kappa$.

• 2. Projection onto the Tangent Space $T_{c}(S_{k})$: Given a vector $w \in \mathbb {R}^{N}$, we need to project w onto $T_{\boldsymbol {c}}(S_{\kappa })$. Since the unit normal to $S_{\kappa }$ at $\boldsymbol {c}$ is $\textbf {u}_{\boldsymbol {c}}$, the projection of w on T_c(S_κ) is given by $w \rightarrow (w - \left \langle w , \textbf {u}_{\boldsymbol {c}} \right \rangle \textbf {u}_{\boldsymbol {c}} )$.

• 3. Covariant Derivative and Integral: Let $\alpha$ be a given path on $S_{\kappa }$, i.e., $\alpha :[0,1] \to S_{\kappa }$, and let w be a vector field along $\alpha$, i.e., for each $\tau \in [0,1]$, $w(\tau ) \in T_{\alpha (\tau )}(S_{\kappa })$. We define the covariant derivative of w along $\alpha$, denoted $\frac {Dw}{d\tau }$, to be the vector field obtained by projecting $\frac {dw}{d\tau }(\tau ) \in \mathbb {R}^{N}$ onto the tangent space $T_{\alpha (\tau )}(S_{\kappa })$. Covariant integral is the inverse procedure of covariant derivative. A vector field u is called a covariant integral of w along $\alpha$ if the covariant derivative of u is w, i.e., $\frac {Du}{d\tau } = w$. Using the previous item on projection, one can derive tools for computing covariant derivatives and integrals of any given vector field.

• 4. Parallel Translation: We will also need tools for forward and backward parallel translation of tangent vectors along a given path $\alpha$ on $S_{\kappa }$. A forward parallel translation of a tangent vector $w \in T_{\alpha (0)}(S_{\kappa })$, is a vector field along $\alpha$, denoted $\tilde {w}$, such that the covariant derivative of $\tilde {w}$ is 0 for all $\tau \in [0,1]$, i.e., $\frac {D \tilde {w}(\tau )}{d\tau } = 0$, and $\tilde {w}(0) = w$. Similarly, backward parallel translation of a tangent vector w ∈ T_α(1)(S_κ), satisfies that $\tilde {w}(1) = w$ and $\frac {D \tilde {w}(\tau )}{d\tau } = 0$ for all $\tau \in [0,1]$.

Algorithm (Path Straightening in $S_{\kappa }$): Given two points $p_{1}$ and $p_{2}$ in $S_{\kappa }$. Suppose $p_{1},p_{2} \in \mathbb {R}^{N}$, and $\tau = 0,1,2,...,k$.

• 1. Initilize a path $\alpha$: for all $\tau = 0,1,2,...k$, using a straight line $(\tau /k)p_{1}+(1-(\tau /k))p_{2}$ in $\mathbb {R}^{N}$. Project each of these points to their nearest points in $S_{\kappa }$ to obtain $\alpha (\tau /k)$.

• 2. Compute $\frac {d \alpha }{d \tau }$ along $\alpha$: let $\tau = 1,2,...,k$ and $v(0) = \textbf {0}$. Compute $v(\tau /k) = k(\alpha (\tau /k)-\alpha ((\tau -1)/k))$ in $\mathbb {R}^{N}$. Project $v(\tau /k)$ into T_α(τ/k)(S_κ) to get $\frac {d \alpha }{dt}(\tau /k)$.

• 3. Compute covariant integral of $\frac {d \alpha }{d\tau }$, with zero initial condition, along $\alpha$ to obtain a vector field u along $\alpha$.

• 4. Backward parallel translate $u(1)$ along $\alpha$ to obtain $\tilde {u}$.

• 5. Compute gradient vector field of E according to $w(\tau /k)=u(\tau /k)-(\tau /k)(\tilde {u}(\tau /k))$ for all $\tau$.

• 6. Update path $\tilde {\alpha }(\tau /k) = \alpha (\tau /k)-\epsilon w(\tau /k)$ by selecting a small $\epsilon >0$. Then project $\tilde {\alpha }(\tau /k)$ to $S_{\kappa }$ to obtain the updated path $\alpha (\tau /k)$.

• 7. Return to step 2 unless $\|w\|$ is small enough or max iteration times reached.