TDA (version 1.9.1)

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 and References.

Usage

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

Value

The function hausdInterval returns a number \(c\). The confidence interval is \([0, c]\).

Arguments

X

an \(n\) by \(d\) matrix of coordinates of sampled points.

m

the size of the subsamples.

B

the number of subsampling iterations. The default value is 30.

alpha

hausdInterval returns a (1-alpha) confidence interval. The default value is 0.05.

parallel

logical: if TRUE, the iterations are parallelized, using the library parallel. The default value is FALSE.

printProgress

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

Author

Fabrizio Lecci

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 (Fasy, Lecci, Rinaldo, Wasserman, Balakrishnan, and Singh, 2013, Theorem 3).

References

Fasy BT, Lecci F, Rinaldo A, Wasserman L, Balakrishnan S, Singh A (2013). "Statistical Inference For Persistent Homology: Confidence Sets for Persistence Diagrams." (arXiv:1303.7117). Annals of Statistics.

See Also

bootstrapBand

Examples

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