Description Usage Arguments Details Value Note Author(s) References See Also

This function calculates the optimal bandwidth matrix (kernel covariance) for a two-dimensional animal tracking dataset, given an autocorrelated movement model (Fleming et al, 2015). This optimal bandwidth can fully take into account all autocorrelation in the data, assuming it is captured by the movement model.

1 2 |

`data` |
2D timeseries telemetry data represented as a |

`CTMM` |
A |

`VMM` |
An optional vertical |

`weights` |
By default, the weights are taken to be uniform, whereas |

`fast` |
Use FFT algorithms for weight optimization. |

`dt` |
Optional lag bin width for the FFT algorithm. |

`precision` |
Fraction of maximum possible digits of precision to target in weight optimization. |

`PC` |
Preconditioner to use: can be "Markov", "circulant", "IID", or "direct". |

`verbose` |
Optionally return the optimal |

`trace` |
Produce tracing information on the progress of weight optimization. |

The `weights=TRUE`

argument can be used to correct temporal sampling bias caused by autocorrelation.
`weights=TRUE`

will optimize `n=length(data$t)`

weights via constrained & preconditioned conjugate gradient algorithms.
These algorithms have a few options that should be considered if the data are very irregular.

`fast=TRUE`

grids the data with grid width `dt`

and applies FFT algorithms, for a computational cost as low as *O(n \log n)* with only *O(n)* function evaluations.
If no `dt`

is specified, the minimum sampling interval `min(diff(data$t))`

is used.
**If the data are irregular (permitting gaps), then dt may need to be several times smaller** to avoid slow down. In this case, try setting

`trace=TRUE`

and decreasing `dt`

until the interations speed up and the number of feasibility assessments becomes less than `dt`

setting will create an excessively high-resolution discretization of time, which will also cause slowdown. In this case `CTMM`

should contain an error model and `dt`

can likely be increased to a larger fraction of the median sampling interval.
`fast=FALSE`

uses exact times and has a computational cost as low as *O(n^2)*, including *O(n^2)* function evaluations. With `PC="direct"`

this method will produce a result that is exact to within machine precision, but with a computational cost of *O(n^3)*.

Returns a bandwidth `matrix`

object, which is to be the optimal covariance matrix of the individual kernels of the kernel density estimate.

To obtain a bandwidth scalar representing the variance of each kernel, a `ctmm`

object with `isotropic=TRUE`

is required. In this case, `bandwidth`

will return bandwidth matrix with identical variances along its diagonal. Note that this will provide an inaccurate estimate for very eccentric distributions.

C. H. Fleming.

T. F. Chan. An Optimal Circulant Preconditioner for Toeplitz Systems. SIAM Journal on Scientific and Statistical Computing, 9:4, 766-771 (1988).

D. Marcotte. Fast variogram computation with FFT. *Computers and Geosciences* 22:10, 1175-1186 (1996).

C. H. Fleming, W. F. Fagan, T. Mueller, K. A. Olson, P. Leimgruber, J. M. Calabrese. Rigorous home-range estimation with movement data: A new autocorrelated kernel-density estimator. Ecology, 96:5, 1182-1188 (2015).

C. H. Fleming, D. Sheldon, W. F. Fagan, P. Leimgruber, T. Mueller, D. Nandintsetseg, M. J. Noonan, K. A. Olson, E. Setyawan, A. Sianipar, J. M. Calabrese. Correcting for missing and irregular data in home-range estimation. Ecological Applications, 28:4, 1003-1010 (2018).

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