TDA (version 1.3)

hausdInterval: Subsampling Confidence Interval for the Hausdorff Distance between a Manifold and a Sample

Description

hausdInterval computes a confidence interval for the Hausdorff distance between a point cloud X and the underlying manifold from which X was sampled. See Details. The validity of the method is proved in the 1st Reference.

Usage

hausdInterval(X, m, B = 30, alpha = 0.05, parallel = FALSE, 
              printProgress=FALSE)

Arguments

X

an $n$ by $d$ matrix of coordinates of sampled points.

m

the size of the subsamples.

B

the number of subsampling iterations.

alpha

hausdInterval returns a (1-alpha) confidence interval.

parallel

logical: if TRUE the iterations are parallelized, using the library parallel.

printProgress

if TRUE a progress bar is printed. Default is FALSE.

Value

• Returns a number $c$. The confidence interval is $[0, c]$.

Details

For B times, the subsampling algorithm subsamples m points of X (without replacement) and computes the Hausdorff distance between the original sample X and the subsample. The result is a sequence of B values. Let $q$ be the (1-alpha) quantile of these values and let $c=2*q$. The interval $[0, c]$ is a valid (1-alpha) confidence interval for the Hausdorff distance between X and the underlying manifold, as proven in Theorem 3 of the first reference.

References

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

bootstrapBand

Examples

X= circleUnif(1000)
interval= hausdInterval(X, m=800)
print(interval)