Computes the bandwidth for lookout from the minimum spanning tree.

This function computes the bandwidth used by lookout() for its kernel density estimates, from the edge lengths of the Euclidean minimum spanning tree of the data.

Usage

find_tda_bw(X, fast = NULL, gamma = 0.98, use_differences = FALSE)

Arguments

X: The numerical input data in a data.frame, matrix or tibble format.
fast: Deprecated and ignored. The bandwidth is always computed using all of X.
gamma: The quantile of the minimum spanning tree edge lengths to return. Default is 0.98. Ignored when use_differences = TRUE.
use_differences: If TRUE, the value returned is the lower end of the largest gap between consecutive minimum spanning tree edge lengths among those at or above their median. If FALSE (the default), the gamma quantile of the edge lengths is returned.

Value

A single number: the bandwidth on the scale of X, before multiplication by sqrt(NCOL(X) + 4).

Details

The Euclidean minimum spanning tree of the rows of X is computed with emst(), giving NROW(X) - 1 edge lengths in increasing order. Each edge length is the distance at which two clusters merge under single-linkage clustering, or equivalently the death radius of a connected component in the Rips filtration of the data, which is why the function is named after topological data analysis.

By default (use_differences = FALSE), the value returned is the gamma quantile of the edge lengths, computed with quantile() using type = 8. With use_differences = TRUE, only the edge lengths at or above their median are kept, and the value returned is the lower of the two consecutive edge lengths with the largest gap between them. This is the rule of Kandanaarachchi and Hyndman (2022), used by lookout() when old_version = TRUE.

The value returned is on the scale of X and is not itself the support radius of the kernel. lookout() multiplies it by sqrt(m + 4), where m = NCOL(X), to obtain the support radius of the Epanechnikov kernel. A spherically symmetric Epanechnikov kernel with support radius h in m dimensions has standard deviation h / sqrt(m + 4) in each coordinate, so the kernel used by lookout() has per-coordinate standard deviation equal to the value returned here. Note that lookout() applies this function to the scaled data when scale = TRUE.

The minimum spanning tree is always computed on all of X.

References

Kandanaarachchi, S, and Hyndman, RJ (2022) Leave-one-out kernel density estimates for outlier detection, J Computational & Graphical Statistics, 31(2), 586-599. https://robjhyndman.com/publications/lookout/.

Hyndman, RJ, Kandanaarachchi, S, and Turner, K (2026) Lookout 2: Anomaly detection via leave-one-out kernel density estimation, arXiv:2603.22636. https://robjhyndman.com/publications/lookout2.html

Examples

X <- rbind(
  data.frame(
    x = rnorm(500),
    y = rnorm(500)
  ),
  data.frame(
    x = rnorm(5, mean = 10, sd = 0.2),
    y = rnorm(5, mean = 10, sd = 0.2)
  )
)
find_tda_bw(X)
#> [1] 0.4605964