TDA (version 1.9.1)

kernelDist: Kernel distance over a Grid of Points

Description

Given a point cloud X, the function kernelDist 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, weight = 1, printProgress = FALSE)

Value

The function 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.

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.

weight

either a number, or a vector of length $n$. If it is a number, then same weight is applied to each points of X. If it is a vector, weight represents weights of each points of X. The default value is 1.

printProgress

if TRUE, a progress bar is printed. The default value is FALSE.

Author

Jisu Kim and Fabrizio Lecci

References

Phillips JM, Wang B, Zheng Y (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)