kernelDist | R Documentation |
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) }.
kernelDist(X, Grid, h, weight = 1, printProgress = FALSE)
X |
an |
Grid |
an |
h |
number: the smoothing paramter of the Gaussian Kernel. |
weight |
either a number, or a vector of length |
printProgress |
if |
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.
Jisu Kim and Fabrizio Lecci
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.
kde
, dtm
, distFct
## 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)
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