For a point datum with a circular normal distribution of position error, the square centred on the datum that a given effort, spread evenly, should cover to make the probability of success as large as possible (Frost 1996, chapter 6). With several efforts, each search is optimized in turn against the distribution as earlier unsuccessful searches left it.
Usage
optsqr(
effort,
model = c("invcube", "random", "definite", "mbeta"),
m = 1,
E = 1
)Arguments
- effort
Positive numeric vector: the effort (sweep width times track length) of each search in turn, in the square of the units of
E.- model
"invcube"(the default; Frost's ideal search conditions),"random"(poor conditions),"definite"or"mbeta".- m
Number in (0, 1]: the M-Beta height. Default 1.
- E
Positive number: the total probable error of position. Default 1.
Value
A data frame with one row per search: search, effort,
releff (effort over E^2), cumeff (cumulative relative
effort), factor, side (2 f E), poc,
coverage, pod, pos and cumpos (the probability
that one of the searches so far succeeded).
Details
The search factor f is the half side of the square in units
of the total probable error of position E (the radius of the circle
holding half the distribution, E = 1.1774 sigma). For relative effort
z = Z / E^2 the coverage is C = z / (2 f)^2, the containment is
the circular normal's mass in the square, (2 Phi(f E / sigma) - 1)^2
for a first search, and the probability of success POC x POD(C) with
the POD curve of model (srchpod). The factor is found on
a grid of 241 values from 0.02 to 10 and refined by optimize(). After
a search the density inside its square is multiplied by 1 - POD; every
later search is optimized against what is left, and its poc and
pos are conditional on the earlier searches having failed. Frost's
figures 6-3 to 6-6 are drawn from it by srchfig.
References
Frost, J. R. (1996). The Theory of Search: A Simplified Explanation. Soza and Company and U.S. Coast Guard Office of Search and Rescue. Chapter 6.
Koopman, B. O. (1957). The theory of search. III. The optimum distribution of searching effort. Operations Research 5(5), 613-626. doi:10.1287/opre.5.5.613
Examples
# Frost's example: an effort of 4 E^2 on a point datum
optsqr(4)
#> ╭── Optimal square searches ────────────────────────────────────────────────╮
#> │ Searches 1 │
#> │ Total effort 4 │
#> │ Chance found by the end 0.4578 │
#> │ │
#> │ search effort releff cumeff factor side poc coverage pod │
#> │ ------ ------ ------ ------ ------ ----- ------ -------- ------ │
#> │ 1 4 4 4 1.021 2.042 0.5940 0.9592 0.7707 │
#> │ │
#> │ search pos cumpos │
#> │ ------ ------ ------ │
#> │ 1 0.4578 0.4578 │
#> │ │
#> │ Each search covers the square (of side 'side') that gives it the best │
#> │ chance of success after the searches before it. poc is the chance the │
#> │ object is in that square, pod the chance of detecting it there, pos │
#> │ their product and cumpos the running chance it has been found. │
#> ╰───────────────────────────────────────────────────────────────────────────╯
# poor search conditions: the random search curve
optsqr(4, model = "random")
#> ╭── Optimal square searches ────────────────────────────────────────────────╮
#> │ Searches 1 │
#> │ Total effort 4 │
#> │ Chance found by the end 0.3664 │
#> │ │
#> │ search effort releff cumeff factor side poc coverage pod │
#> │ ------ ------ ------ ------ ------ ----- ------ -------- ------ │
#> │ 1 4 4 4 1.028 2.055 0.5986 0.9470 0.6121 │
#> │ │
#> │ search pos cumpos │
#> │ ------ ------ ------ │
#> │ 1 0.3664 0.3664 │
#> │ │
#> │ Each search covers the square (of side 'side') that gives it the best │
#> │ chance of success after the searches before it. poc is the chance the │
#> │ object is in that square, pod the chance of detecting it there, pos │
#> │ their product and cumpos the running chance it has been found. │
#> ╰───────────────────────────────────────────────────────────────────────────╯
# three searches in a row, each optimal for what the last ones left
optsqr(c(4, 4, 8), E = 2)
#> ╭── Optimal square searches ────────────────────────────────────────────────╮
#> │ Searches 3 │
#> │ Total effort 16 │
#> │ Chance found by the end 0.419 │
#> │ │
#> │ search effort releff cumeff factor side poc coverage pod │
#> │ ------ ------ ------ ------ ------ ----- ------ -------- ------ │
#> │ 1 4 1 1 0.6432 2.573 0.3038 0.6043 0.5511 │
#> │ 2 4 1 2 0.9360 3.744 0.4382 0.2854 0.2794 │
#> │ 3 8 2 4 1.1183 4.473 0.5339 0.3998 0.3837 │
#> │ │
#> │ search pos cumpos │
#> │ ------ ------ ------ │
#> │ 1 0.1674 0.1674 │
#> │ 2 0.1224 0.2694 │
#> │ 3 0.2049 0.4190 │
#> │ │
#> │ Each search covers the square (of side 'side') that gives it the best │
#> │ chance of success after the searches before it. poc is the chance the │
#> │ object is in that square, pod the chance of detecting it there, pos │
#> │ their product and cumpos the running chance it has been found. │
#> ╰───────────────────────────────────────────────────────────────────────────╯
try(optsqr(-1))
#> Error : optsqr: `effort` must be positive