overlap: Calculate the overlap between two stationary distributions

In ctmm-initiative/ctmm: Continuous-Time Movement Modeling

Calculate the overlap between two stationary distributions

Description

This function calculates a useful measure of similarity between distributions known as the Bhattacharyya coefficient in statistics and simply the fidelity or overlap in quantum and statistical mechanics. It is roughly speaking the ratio of the intersection area to the average individual area, but it is a direct comparison between the density functions and does not require an arbitrary quantile to be specified. When applied to ctmm objects, this function returns the overlap of the two Gaussian distributions. When applied to aligned UD objects with corresponding movement models, this function returns the overlap of their (autocorrelated) kernel density estimates.

Usage

 overlap(object,method="Bhattacharyya",level=0.95,debias=TRUE,...) 

Arguments

object	A list of ctmm fit or aligned UD objects to compare.
method	Can be "Bhattacharyya" or "Encounter" (see Details below).
level	The confidence level desired for the output.
debias	Approximate debiasing of the overlap.
...	Not currently used.

Details

The default method="Bhattacharyya" estimates the standard overlap measure \int\int \sqrt{p(x,y) \, q(x,y)} \, dx \, dy between the distributions p(x,y) and q(x,y), while method="encounter" estimates the non-standard measure \frac{\int\int p(x,y) \, q(x,y) \, dx \, dy}{\sqrt{\int\int p(x',y')^2 \, dx' dy' \int\int q(x'',y'')^2 \, dx'' dy''}}, which has a numerator proportional to the uncorrelated encounter probability and UD overlap index (Tilberg and Dixon, 2022). Both measures lie between 0 and 1, where 0 indicates no shared support and 1 indicates identical distributions.

Value

An object with slots DOF, containing the effective sample sizes, and CI containing a table of confidence intervals on the overlap estimates. A value of 1 implies that the two distributions are identical, while a value of 0 implies that the two distributions share no area in common.

Note

In ctmm v0.5.2, direct support for telemetry objects was dropped and the CTMM argument was depreciated for UD objects, simplifying usage.

Uncertainties in the model fits are propagated into the overlap estimate under the approximation that the Bhattacharyya distance is a chi-square random variable. Debiasing makes further approximations noted in Winner & Noonan et al (2018).

Author(s)

C. H. Fleming and K. Winner

References

K. Winner, M. J. Noonan, C. H. Fleming, K. Olson, T. Mueller, D. Sheldon, J. M. Calabrese. “Statistical inference for home range overlap”, Methods in Ecology and Evolution, 9:7, 1679-1691 (2018) \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1111/2041-210X.13027")}.

M. Tilberg, P. M. Dixon, “Statistical inference for the utilization distribution overlap index (UDOI)”, Methods in Ecology and Evolution, 13:5, 1082-1092 (2022) \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1111/2041-210X.13813")}.

See Also

akde, ctmm.fit, distance, encounter

Examples

# Load package and data
library(ctmm)
data(buffalo)

# fit models for first two buffalo
GUESS <- lapply(buffalo[1:2], function(b) ctmm.guess(b,interactive=FALSE) )
# using ctmm.fit here for speed, but you should almost always use ctmm.select
FITS <- lapply(1:2, function(i) ctmm.fit(buffalo[[i]],GUESS[[i]]) )
names(FITS) <- names(buffalo[1:2])

# Gaussian overlap between these two buffalo
overlap(FITS)

# AKDE overlap between these two buffalo
# create aligned UDs
UDS <- akde(buffalo[1:2],FITS)
# evaluate overlap
overlap(UDS)

ctmm-initiative/ctmm documentation built on April 18, 2024, 9:39 a.m.