Correlated random walk of a series of events

Treats the successive locations of a series (incidents attributed to one offender, in time order) as a correlated random walk (Kareiva and Shigesada 1983): steps of independent lengths and changes of heading (turning angles) independent of each other and of the steps. The fit gives the mean and mean squared step and the mean cosine and sine of the turns; the expected squared displacement after k moves is Kareiva and Shigesada's for symmetric turning (mean sine 0), k E l^2 + 2 (E l)^2 c / (1 - c) (k - (1 - c^k) / (1 - c)), next to the observed mean over every start. The walk is then continued nsim times for h steps from the last location and heading, each turn and step drawn independently and uniformly from the observed ones (an empirical correlated random walk), and the simulated positions summarise where the next events may fall. It describes movement; it does not identify anyone, and a series with few events gives a rough walk.

Usage

crwalk(x, y = NULL, h = 1, nsim = 999, seed = NULL, lonlat = FALSE)

Arguments

x, y

The points: x and y numeric vectors of equal length; or x a matrix or data frame with columns named x and y (or X and Y, as sf::st_coordinates() names them, or lon and lat with lonlat = TRUE) and y NULL; or x an sf or sfc object of POINT geometries, whose coordinate reference system must agree with lonlat. An unnamed two-column matrix is read as x then y; other columns are never taken by position. Vectors given separately (x, y, and w or t where a function takes them) are matched by name when every one is named and no name repeats, as elsewhere in the package; otherwise by position. No missing or infinite values.

h

The number of steps ahead: a whole number from 1.

nsim

The number of simulated continuations: a whole number from 1.

seed

NULL to draw the simulations' seed from R's generator (so set.seed() fixes them), or one whole number from 0.

lonlat

FALSE (the default) when x and y are planar coordinates in one length unit; TRUE when they are longitude and latitude in degrees, which are then projected to metres by eqproj() about their own centre, and centres are returned in degrees. It is never guessed.

Value

A list. fit, a named vector: steps, ml (mean step), ml2 (mean squared step), c and s (mean cosine and sine of the turning angles) and head (the last heading, radians anticlockwise from east). msd, a data frame with one row per step ahead k: obs (the observed mean squared displacement over k moves, NA beyond the series), exp (its expectation; it assumes s near 0), and x, y, sdx, sdy (the mean and standard deviations of the simulated positions; the means in degrees when lonlat = TRUE, the rest in metres). sim, the simulated positions: matrices x and y, one row per walk and one column per step (planar, metres when lonlat = TRUE).

References

Kareiva, P. M. and Shigesada, N. (1983). Analyzing insect movement as a correlated random walk. Oecologia, 56(2-3), 234-238. doi:10.1007/BF00379695

Examples

set.seed(1)
a <- cumsum(rnorm(20, 0, 0.5))
p <- cbind(x = cumsum(cos(a)), y = cumsum(sin(a)))
r <- crwalk(p, h = 3, nsim = 200, seed = 1)
r$fit
#>      steps         ml        ml2          c          s       head 
#> 19.0000000  1.0000000  1.0000000  0.8935287  0.1187582  1.9052388 
r$msd
#>   k      obs      exp        x        y       sdx       sdy
#> 1 1 1.000000 1.000000 12.12662 10.09144 0.3882481 0.1794948
#> 2 2 3.787057 3.787057 11.60345 10.74197 0.7778394 0.4737584
#> 3 3 8.165410 8.170902 11.07326 11.25329 1.1794886 0.8584427