Calculates up to three estimates of activity pattern overlap based on times of observations for two species.

1 | ```
overlapEst(A, B, kmax = 3, adjust=c(0.8, 1, 4), n.grid = 128)
``` |

`A` |
a vector of times of observations of species A in radians, ie. scaled to [0, |

`B` |
a vector of times of observations of species B in radians. |

`kmax` |
maximum value of k for optimal bandwidth estimation. |

`adjust` |
bandwidth adjustment; either a single value used for all 3 overlap estimates, or a vector of 3 different values; a NA value in |

`n.grid` |
number of points at which to estimate density for comparison between species; smaller values give lower precision but run faster in simulations and bootstraps. |

See `overlapTrue`

for the meaning of coefficient of overlapping, *Δ*.

These estimators of *Δ* use kernel density estimates fitted to the data to approximate the true density functions *f(t)* and *g(t)*. Schmid & Schmidt (2006) propose five estimators of overlap:

Dhat1 is calculated from vectors of densities estimated at *T* equally-spaced times, *t*, between 0 and *2π*:

For circular distributions, Dhat2 is equivalent to Dhat1, and Dhat3 is inapplicable.

Dhat4 and Dhat5 use vectors of densities estimated at the times of the observations of the species, *x* and *y*:

where *n, m* are the sample sizes and *I* is the indicator function (1 if the condition is true, 0 otherwise).

Returns a named vector of three estimates of overlap. Individual elements may be NA if the argument `adjust`

contained NAs. All will be NA if optimal bandwidth estimation failed.

Mike Meredith, based on work by Martin Ridout.

Ridout & Linkie (2009) Estimating overlap of daily activity patterns from camera trap data. *Journal of Agricultural, Biological, and Environmental Statistics* 14:322-337

Schmid & Schmidt (2006) Nonparametric estimation of the coefficient of overlapping - theory and empirical application, *Computational Statistics and Data Analysis*, 50:1583-1596.

1 2 3 4 5 6 7 8 9 10 11 | ```
# Get example data:
data(simulatedData)
# Use defaults:
overlapEst(tigerObs, pigObs)
# Dhat1 Dhat4 Dhat5
# 0.2908618 0.2692011 0.2275000
overlapEst(tigerObs, pigObs, adjust=c(NA, 1, NA))
# Dhat1 Dhat4 Dhat5
# NA 0.2692011 NA
``` |

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