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In fieldopt: Area-Frame and Dual-Frame Survey Design under Real Field Cost

A survey that goes to the field has two costs that textbooks keep apart: the statistical cost, the variance of the estimator for a given sample size, and the field cost, the travel between units and the interviews, which is what the budget actually pays for. fieldopt puts the two side by side. This vignette walks through one design from the frame to the cost-variance frontier.

A frame with a depot

The frame is a data frame with one row per sampling unit (a segment, a farm, a census block) and its coordinates. One row is the depot, the office the teams leave from and return to. Here the frame is synthetic: forty segments in a 10 x 10 km square, with a crop area that trends from west to east.

library(fieldopt)
library(dplyr, warn.conflicts = FALSE)
set.seed(2026)
frame <- tibble(
  unit = c("depot", paste0("s", 1:40)),
  x = c(5, runif(40, 0, 10)), y = c(5, runif(40, 0, 10))
) |>
  mutate(crop = if_else(unit == "depot", NA_real_, 20 + 3 * x + rnorm(n(), sd = 2)))
frame
#> # A tibble: 41 × 4
#>    unit      x     y  crop
#>    <chr> <dbl> <dbl> <dbl>
#>  1 depot 5     5      NA  
#>  2 s1    6.99  3.71   42.5
#>  3 s2    5.57  8.75   34.3
#>  4 s3    1.40  4.35   24.8
#>  5 s4    2.86  4.40   26.9
#>  6 s5    5.55  0.820  39.5
#>  7 s6    0.251 0.346  22.2
#>  8 s7    4.66  9.29   33.2
#>  9 s8    8.61  2.95   47.4
#> 10 s9    2.53  5.19   28.4
#> # ℹ 31 more rows

A real frame usually starts as a spatial object: as_frame() turns an sf layer of segments or points into this data frame (centroids in latitude and longitude, the area of each polygon), and as_sf() puts samples and results back on the geometries for maps.

Travel and cost

travel_matrix() computes the matrix of travel distances between all units (or, given a sample and the depot as bases, only between the depot and the sampled units, which is what the routing needs). With longitude and latitude it uses great-circle distances in kilometres; with planar coordinates, as here, Euclidean distances. A matrix computed on a road network (for example with OSRM) can be passed through the matrix argument and is the better input when it exists.

m <- travel_matrix(frame, method = "euclidean")
m
#> Travel matrix: 41 units, euclidean (distance); mean off-diagonal 4.886.

field_cost_model() turns a routing solution into money: a cost per unit of travel, a fixed cost per visited unit (access, setting up) and a cost per interview.

model <- field_cost_model(
  per_travel = 2, per_unit = 30, per_interview = 10,
  interviews_per_unit = 4, currency = "BRL"
)
model
#> Field cost model (BRL): 2 per travel unit, 0 per route, 30 per visited unit, 10
#> per interview.

Select, route, estimate

select_units() draws a spatially balanced probability sample with the local pivotal method: the inclusion probabilities are known, and units close to each other are rarely selected together. Spatial balance usually lowers the variance of totals of spatially structured variables; it also spreads the units over the territory, which is exactly what makes the field work longer. Making that tension explicit is the point of the package.

s <- frame |>
  filter(unit != "depot") |>
  select_units(n = 12)
s |>
  sampled() |>
  select(unit, x, y, pi)
#> # A tibble: 12 × 4
#>    unit       x     y    pi
#>    <chr>  <dbl> <dbl> <dbl>
#>  1 s3    1.40   4.35    0.3
#>  2 s6    0.251  0.346   0.3
#>  3 s8    8.61   2.95    0.3
#>  4 s11   0.0593 9.72    0.3
#>  5 s12   6.92   6.33    0.3
#>  6 s15   1.54   5.23    0.3
#>  7 s21   3.42   2.98    0.3
#>  8 s22   3.43   8.96    0.3
#>  9 s27   4.11   6.30    0.3
#> 10 s35   5.86   1.86    0.3
#> 11 s36   4.26   6.33    0.3
#> 12 s40   4.47   9.00    0.3

route_fieldwork() routes the field work from the depot through the selected units. Without limits it returns one tour; with a maximum length or a maximum number of stops per route, it splits the work into routes, one per team or per day. Small instances are solved exactly; larger ones by a hybrid genetic search with an optimal split of the tour and a local search, with a lower bound and, for single tours, a proof of optimality, so that the quality of the solution is visible. The sample can be piped straight into the routing.

r <- s |>
  travel_matrix(bases = frame |> filter(unit == "depot"), method = "euclidean") |>
  route_fieldwork(units = s, depot = "depot", max_stops = 5, cost_model = model)
r
#> 
#> ── Field routes ────────────────────────────────────────────────────────────────
#> 3 routes from "depot" through 12 units: total travel 42.62 distance (proven
#> optimal).
#> Route 1 (17.81): s3 > s15 > s11 > s22 > s40
#> Route 2 (21.56): s21 > s6 > s35 > s8 > s12
#> Route 3 (3.254): s36 > s27
#> Cost (BRL): travel 85.24 + routes 0 + units 360 + interviews 480 = 925.2.
autoplot(r)

Figure of the chunk route

design_variance() gives the Horvitz-Thompson estimate of the total of crop and its variance by the local-mean estimator of Grafström and Schelin (2014), which suits spatially balanced samples and needs no joint inclusion probabilities. The variance the same sample size would give under simple random sampling is shown for comparison.

design_variance(s, y = "crop")
#> # A tibble: 1 × 9
#>   total variance    se     cv variance_srs     n     N variance_method    df
#>   <dbl>    <dbl> <dbl>  <dbl>        <dbl> <int> <int> <chr>           <int>
#> 1 1254.    3158.  56.2 0.0448        5874.    12    40 local-mean         11
sum(frame$crop, na.rm = TRUE)
#> [1] 1241.484

The cost-variance frontier

A design is chosen by its sample size, its routing limits and its cost model. cost_variance_frontier() repeats the select-route-price-estimate cycle for a grid of sample sizes and returns the mean cost and the mean variance at each size: the frontier a survey planner faces.

f <- cost_variance_frontier(
  frame,
  depot = "depot", cost_model = model,
  n_grid = c(6, 9, 12, 16, 20, 24), y = "crop",
  method = "euclidean", max_stops = 5,
  n_rep = 8, iterations = 60
)
f
#> # A tibble: 6 × 9
#>       n cost_mean cost_sd cost_per_unit travel_mean routes_mean variance_mean
#> * <dbl>     <dbl>   <dbl>         <dbl>       <dbl>       <dbl>         <dbl>
#> 1     6      475.   6.08           79.2        27.6           2        13594.
#> 2     9      701.   4.64           77.9        35.5           2         4791.
#> 3    12      925.   8.47           77.1        42.7           3         2958.
#> 4    16     1218.   6.22           76.1        48.9           4         1529.
#> 5    20     1517.   0.949          75.9        58.7           4         1136.
#> 6    24     1810.   3.69           75.4        65.1           5          825.
#> # ℹ 2 more variables: cv_mean <dbl>, n_rep <dbl>
autoplot(f)

Figure of the chunk frontier

The frontier makes two questions answerable with numbers: how much the next unit of precision costs in the field, and how much a different field organisation (more teams, longer days, a closer depot) would change that price. The vignettes routing-teams and frames-and-allocation take those questions up.

References

Cochran, W. G. (1977). Sampling Techniques, third edition. Wiley.

Grafström, A., Lundström, N. L. P. and Schelin, L. (2012). Spatially balanced sampling through the pivotal method. Biometrics, 68(2), 514--520.

Grafström, A. and Schelin, L. (2014). How to select representative samples. Scandinavian Journal of Statistics, 41(2), 277--290.

Prins, C. (2004). A simple and effective evolutionary algorithm for the vehicle routing problem. Computers & Operations Research, 31(12), 1985--2002.