TDA (version 1.6.2)

kde: Kernel Density Estimator over a Grid of Points

Description

Given a point cloud X ($n$ points), the function kde computes the Kernel Density Estimator over a grid of points. The kernel is a Gaussian Kernel with smoothing parameter h. For each $x\in R^d$, the Kernel Density estimator is defined as $$p_X(x)=\frac{1}{n(\sqrt{2\pi }h{)}^d}\sum _{i=1}^n\exp \left(\frac{-\Vert x-X_i{\Vert }_2^2}{2h^2}\right).$$

Usage

kde(X, Grid, h, kertype = "Gaussian", weight = 1,
    printProgress = FALSE)

Arguments

X

an $n$ by $d$ matrix of coordinates of points used in the kernel density estimation process, 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.

kertype

string: if kertype = "Gaussian", Gaussian kernel is used, and if kertype = "Epanechnikov", Epanechnikov kernel is used. Defaults to "Gaussian".

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.

Value

The function kde returns a vector of length $m$ (the number of points in the grid) containing the value of the kernel density estimator for each point in the grid.

References

Larry Wasserman (2004), "All of statistics: a concise course in statistical inference", Springer.

Brittany T. Fasy, Fabrizio Lecci, Alessandro Rinaldo, Larry Wasserman, Sivaraman Balakrishnan, and Aarti Singh. (2013), "Statistical Inference For Persistent Homology: Confidence Sets for Persistence Diagrams", (arXiv:1303.7117). To appear, Annals of Statistics.

See Also

kernelDist, distFct, dtm

Examples

# NOT RUN {
## Generate Data from the unit circle
n <- 300
X <- circleUnif(n)

## Construct a grid of points over which we evaluate the function
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 density estimator
h <- 0.3
KDE <- kde(X, Grid, h)
# }