TDA (version 1.0)

kernelDist: Kernel distance over a Grid of Points

Description

Given a point cloud X, this function computes the kernel distance over a grid of points. The kernel is a Gaussian Kernel with smoothing parameter h: $$K_h(x,y)=\exp\left( \frac{- \Vert x-y \Vert_2^2}{2h^2} \right).$$ For each $x \in R^d$ the Kernel distance is defined by $$\kappa_X(x)=\sqrt{ \frac{1}{n^2} \sum_{i=1}^n\sum_{j=1}^n K_h(X_i, X_j) + K_h(x,x) - 2 \frac{1}{n} \sum_{i=1}^n K_h(x,X_i) }.$$

Usage

kernelDist(X, Grid, h)

Arguments

X

an $n$ by $d$ matrix of coordinates of points, where $n$ is the number of points and $d$ is the dimension.

Grid

an $m$ by $d$ matrix of coordinates, where $m$ is the number of points in the grid.

h

number: the smoothing paramter of the Gaussian Kernel.

Value

• kernelDist returns a vector of lenght $m$ (the number of points in the grid) containing the value of the Kernel distance for each point in the grid.

References

Jeff M. Phillips, Bei Wang, and Yan Zheng (2013), "Geometric Inference on Kernel Density Estimates," arXiv:1307.7760.

Chazal F, Fasy BT, Lecci F, Michel B, Rinaldo A, Wasserman L (2014). "Robust Topological Inference: Distance-To-a-Measure and Kernel Distance." Technical Report.

See Also

kde, dtm, distFct

Examples

## Generate Data from the unit circle
n = 300
X = circleUnif(n)

## Construct a grid of points over which we evaluate the functions
by=0.065
Xseq=seq(-1.6, 1.6, by=by)
Yseq=seq(-1.7, 1.7, by=by)
Grid=expand.grid(Xseq,Yseq)

## kernel distance estimator
h=0.3
Kdist= kernelDist(X, Grid, h)