Fixed-radius concentration can be formulated as a weighted circle-placement problem: given point locations with associated weights and a disk of fixed radius, find the centre of the disk that maximises the total weight enclosed. This is a classical spatial optimisation problem from computational geometry.
In its simplest form, the question is: where can I place a 200 metre circle so that the total insured value inside that circle is as large as possible?
In an exposure portfolio, the point weights might represent insured value, population, asset value, infrastructure value, or another quantity observed at point locations. Insurance concentration analysis is one practical application: the same fixed-radius problem can be interpreted as finding the largest local accumulation of insured value.
The package does not impose a probabilistic model. It computes deterministic spatial aggregates from observed point locations and values. The workflows and parameter choices shown in this vignette are illustrative examples of the computational building blocks, not a prescribed methodology or process for any particular organisation.
The examples below use the included Groningen data. The
same functions can be applied to larger portfolios.
library(spatialrisk)
portfolio <- Groningen
portfolio <- portfolio[, c("lon", "lat", "amount")]
head(portfolio)
#> # A tibble: 6 × 3
#> lon lat amount
#> <dbl> <dbl> <dbl>
#> 1 6.57 53.2 24
#> 2 6.55 53.2 33
#> 3 6.57 53.2 48
#> 4 6.56 53.2 7
#> 5 6.57 53.2 16
#> 6 6.56 53.2 28The amount column is the value to be accumulated within
each radius. In an insurance application this could represent an insured
amount, exposure measure, or another portfolio value. The 200 metre
radius used below is illustrative; the relevant radius depends on the
analytical or reporting context.
The main applied workflow is concentration_hotspot(). It
takes point-level exposure data, a value column, and a radius, and
returns the circle centre with the largest fixed-radius sum.
hotspot <- concentration_hotspot(
portfolio,
value = "amount",
radius = 200,
cell_size = 100,
progress = FALSE
)
hotspot
#> <hotspot>
#> Number of hotspots: 1
#> Radius: 200 meters
#> Value: amount
#>
#> id lon lat amount_sum
#> 1 1 6.547318 53.23659 64438The reported amount_sum is the sum of
amount within 200 metres of the selected centre. The
default method = "continuous" searches for a centre that
may lie anywhere in space, not only on an observed building
location.
A hotspot result contains two main components:
hotspots: the selected centre coordinates and the
fixed-radius sum;contributing_points: the observations inside the
selected hotspot radius.hotspot$hotspots
#> id lon lat amount_sum
#> 1 1 6.547318 53.23659 64438
head(hotspot$contributing_points[, c("id", "data_row", "lon", "lat",
"amount", "amount_sum")])
#> id data_row lon lat amount amount_sum
#> 1 1 1492 6.545297 53.23569 148 64438
#> 2 1 4703 6.545482 53.23547 132 64438
#> 3 1 18287 6.545429 53.23546 130 64438
#> 4 1 19958 6.545392 53.23543 138 64438
#> 5 1 22587 6.545493 53.23545 142 64438
#> 6 1 19 6.544724 53.23646 411 64438This separation between the hotspot centre and the contributing
observations is important in applied insurance work. It allows the
result to be inspected, mapped, and reconciled with the underlying
portfolio. The data_row column identifies the row in the
original input data.
The observations assigned to the selected hotspot are stored directly in the hotspot object. This is the safest way to audit the reported hotspot, because it uses the exact contributing observations returned by the search.
head(hotspot$contributing_points)
#> id data_row lon lat amount distance_m amount_sum
#> 1 1 1492 6.545297 53.23569 148 168.0422 64438
#> 2 1 4703 6.545482 53.23547 132 174.7968 64438
#> 3 1 18287 6.545429 53.23546 130 177.9221 64438
#> 4 1 19958 6.545392 53.23543 138 182.1778 64438
#> 5 1 22587 6.545493 53.23545 142 175.8840 64438
#> 6 1 19 6.544724 53.23646 411 173.7389 64438
nrow(hotspot$contributing_points)
#> [1] 208
sum(hotspot$contributing_points$amount)
#> [1] 64438The lower-level function points_within_radius() applies
the same local inclusion rule around a specified centre. It is useful
for checking a known or externally specified location.
known_centre_points <- points_within_radius(
portfolio,
lon_center = 6.5549,
lat_center = 53.1942,
radius = 200
)
head(known_centre_points)
#> # A tibble: 6 × 4
#> lon lat amount distance_m
#> <dbl> <dbl> <dbl> <dbl>
#> 1 6.56 53.2 523 57.5
#> 2 6.56 53.2 513 64.5
#> 3 6.56 53.2 515 70.3
#> 4 6.55 53.2 246 70.6
#> 5 6.56 53.2 768 76.6
#> 6 6.55 53.2 238 79.2
nrow(known_centre_points)
#> [1] 110
sum(known_centre_points$amount)
#> [1] 25668The returned data contains the observations that contribute to this specified local fixed-radius sum. This makes the aggregate traceable back to the underlying policies or locations.
The same operation can be repeated for several target locations with
radius_sum(). Here, the target locations are the first five
observations in the portfolio.
targets <- portfolio[1:5, c("lon", "lat")]
target_sums <- radius_sum(
targets = targets,
reference = portfolio,
value = "amount",
radius = 200,
progress = FALSE,
result_col = "amount_200m"
)
target_sums
#> # A tibble: 5 × 3
#> lon lat amount_200m
#> <dbl> <dbl> <dbl>
#> 1 6.57 53.2 8612
#> 2 6.55 53.2 16704
#> 3 6.57 53.2 9120
#> 4 6.56 53.2 7970
#> 5 6.57 53.2 8633The targets and reference arguments are
separated deliberately. This makes it possible to evaluate concentration
at existing policy locations, at externally specified coordinates, or at
candidate centres created in a custom workflow. In this sense,
points_within_radius() and radius_sum() are
supporting functions: they provide transparency and flexibility around
the main hotspot search.
The maximum fixed-radius circle does not generally need to be centred on one of the insured locations. This distinction matters because a circle placed between several buildings may cover a larger total value than any circle centred exactly on a building.
spatialrisk therefore distinguishes two useful search
strategies:
method = "continuous": the centre may lie anywhere in
space;method = "observed": candidate centres are limited to
observed point locations.The observed-points method is fast and deterministic, and is useful as a benchmark. It can, however, miss a higher concentration when the best circle centre lies between buildings.
hotspot_continuous <- concentration_hotspot(
portfolio,
value = "amount",
radius = 200,
cell_size = 100,
progress = FALSE
)
hotspot_observed <- concentration_hotspot(
portfolio,
value = "amount",
radius = 200,
method = "observed",
progress = FALSE
)
rbind(
continuous = hotspot_continuous$hotspots,
observed = hotspot_observed$hotspots
)
#> id lon lat amount_sum
#> continuous 1 6.547318 53.23659 64438
#> observed 1 6.547288 53.23664 64172In this example the continuous hotspot has a higher
amount_sum than the observed-points hotspot. The example
should be read as a demonstration of the methodological distinction:
restricting centres to observed locations changes the optimisation
problem.
The original grid-refinement workflow remains available with
method = "grid". In that method, grid_spacing
controls the local grid refinement. For the default
method = "continuous", grid_spacing is only
used if the local pair-refinement subset is too large and the function
falls back to grid refinement.
The argument n_hotspots gives the number of hotspots to
return. When n_hotspots > 1, hotspots are selected
sequentially: after the first hotspot has been found, its contributing
observations are removed before the next hotspot is searched for. This
gives non-overlapping hotspot assignments.
hotspot_top2 <- concentration_hotspot(
portfolio,
value = "amount",
radius = 200,
cell_size = 100,
progress = FALSE,
n_hotspots = 2
)
hotspot_top2$hotspots
#> id lon lat amount_sum
#> 1 1 6.547331 53.23659 64438
#> 2 2 6.523439 53.23094 57977The first hotspot addresses the single-circle maximum concentration problem. The second hotspot is the largest hotspot in the remaining portfolio after removing the observations assigned to the first hotspot. This greedy procedure is useful for reporting several distinct local accumulations, but it should not be interpreted as a globally optimal joint placement of several circles.
The high-level hotspot workflow can also be run step by step. This is useful when the intermediate candidate selection needs to be inspected before the final hotspot is optimised.
The state supplied to optimize_hotspot() determines
where geometric candidate centres are generated. A prepared state has
not been screened and therefore represents the full active portfolio as
the candidate-generation universe. A state returned by
select_candidates() represents the screened candidate
regions. In either case, retained candidate centres are evaluated
against the complete active portfolio: candidate selection does not
create a subportfolio for scoring.
Conceptually, the search has four stages:
model <- prepare_spatialrisk(portfolio, value = "amount", radius = 200,
cell_size = 100)
model <- select_candidates(model, progress = FALSE)
step_hotspot <- optimize_hotspot(model, n_hotspots = 2, progress = FALSE)
step_hotspot$hotspots
#> id lon lat amount_sum
#> 1 1 6.547331 53.23659 64438
#> 2 2 6.523439 53.23094 57977For small validation problems, candidate screening can be omitted
deliberately. The direct route evaluates observed centres and the valid
pairwise circle-intersection centres generated from the complete active
portfolio. The screened route is the normal continuous search used by
concentration_hotspot().
validation_portfolio <- portfolio[1:200, ]
validation_model <- prepare_spatialrisk(
validation_portfolio,
value = "amount",
radius = 200,
cell_size = 100
)
# Full geometric reference search
full <- optimize_hotspot(validation_model, progress = FALSE)
# Screened continuous search
screened <- validation_model |>
select_candidates(progress = FALSE) |>
optimize_hotspot(progress = FALSE)
full$hotspots
#> id lon lat amount_sum
#> 1 1 6.554816 53.19424 1315
screened$hotspots
#> id lon lat amount_sum
#> 1 1 6.558472 53.19492 1315The full route has pairwise computational cost and is intended for
small portfolios, diagnostics, and methodological validation. It does
not silently switch to grid refinement when the number of points exceeds
max_refinement_points; instead, it warns that the requested
complete search may be expensive.
Calling plot(model) after
prepare_spatialrisk() but before candidate selection shows
the rasterised portfolio sum per cell. After
select_candidates(), plot(model) shows the
centre cells retained by screening. The cell colours still show the
original focal sums;
model$candidates$cells$point_upper_bound contains the
tighter point-based bounds for the default continuous search.
prepared <- prepare_spatialrisk(portfolio, value = "amount", radius = 200,
cell_size = 100)
plot(prepared)The lower bound can also be supplied explicitly, for example
select_candidates(model, threshold = 1000). When
threshold = NULL, the automatic lower bound is deliberately
conservative. The function takes the five highest cells from the focal
raster and evaluates a 10 by 10 regular trial grid across each cell.
These 500 evaluations use the same projected Euclidean geometry and
complete active portfolio as the final search; trial coordinates on
shared cell edges need not be unique. The best feasible concentration
becomes the lower bound. For continuous search with non-negative values,
focal cells above this lower bound undergo a second, point-based bound
calculation described below. The five highest surviving point bounds
seed another trial grid; the best actually attained portfolio sum can
raise, but never lower, the threshold. Only cells whose point bound is
at least the resulting threshold remain candidates. The candidate map is
therefore an inspection view of where the next hotspot may be found, not
a fixed list of final hotspots.
When n_hotspots > 1, the search is repeated. After
the first hotspot has been found, its contributing observations are
removed from the remaining portfolio and the screening, candidate
selection, and refinement steps are run again for the next hotspot. This
is why a candidate map that currently shows, for example, five focal
cells can still lead to ten hotspots when
optimize_hotspot(model, n_hotspots = 10) is used: the five
cells describe the first search iteration only.
The default continuous method uses terra rasterisation and focal sums
for the screening step. It then refines all candidate areas above the
lower bound by evaluating observed local points and the circle centres
implied by local point pairs. The point-to-cell assignment created
during preparation is reused to retrieve points from nearby raster
cells, rather than scanning the complete portfolio separately for every
focal candidate cell. For the common single-hotspot search, the retained
local point sets are combined and Rcpp traverses the pair geometry once
as a streaming angular sweep. The sweep maintains the exact
active-portfolio total at pair-intersection events, without
materialising all centres or running a separate full radius query for
each one. Only centres that can match or improve the current best
receive a confirming indexed radius evaluation. Runtime therefore
depends strongly on local pair density, not only on the total portfolio
size. If the local refinement subset is larger than
max_refinement_points, the function falls back to grid
refinement for that iteration.
The focal moving window is deliberately wider than the requested radius by one raster-cell diagonal. Consequently, with non-negative values, the focal sum is an upper bound for the exact radius sum of every centre located in that cell: all source cells that could contribute to such a centre are included. The automatically estimated threshold is a lower bound obtained from a feasible preliminary centre. A raster cell whose focal upper bound is below that threshold cannot contain a better centre and need not be refined.
For each surviving centre cell \(C\), a tighter bound is
\[ V(C) = \sum_i w_i\,\mathbf{1}\{d(p_i,C)\leq r\}, \]
where \(d(p_i,C)\) is the minimum distance from the point to the closed cell rectangle. If a point can lie within a radius-\(r\) disk centred anywhere in \(C\), it must contribute to this bound. The reverse need not hold for all points simultaneously, so \(V(C)\) remains an upper bound, not an attainable hotspot value. Unlike the focal sum, it does not include the full weight of a neighbouring cell merely because its boundary can be reached. The implementation uses the stored terra cell assignments to retrieve nearby active points in one batch and allows for radius-scoring and floating-point tolerances at cell edges. Cells with \(V(C)<L\) are removed; equality is retained. The feasible lower bound \(L\) is strengthened using actual disk evaluations, never by substituting an upper bound. This additional screening changes only the possible centre domain, not the active portfolio used to evaluate retained centres.
For the remaining local point pairs, both radius-circle intersection centres are constructed geometrically. Each of those two centres is assigned to a raster cell using terra, and the centres are screened separately. Only a centre whose own cell passed focal screening is considered by the exact angular evaluation. This is more selective than retaining both centres merely because one of them lies in a candidate cell. Every retained centre is evaluated using the full remaining active portfolio, not only the local points used to generate it. Points outside the candidate-generation subset therefore still contribute whenever they lie within the radius.
This additional centre-level pruning is used only with non-negative
values and the default automatically estimated lower bound. For a
user-supplied threshold or negative values, the implementation does not
rely on this upper-bound argument and uses the broader refinement route.
The direct validation route
optimize_hotspot(prepare_spatialrisk(...)) also remains
unfiltered and evaluates the complete geometric candidate set.
Fixed-radius concentration is a point-level calculation. For
communication and reporting, it is often useful to summarise values by
administrative or portfolio regions. The function
summarise_points_by_polygon() joins point data to polygons
and applies a summary function.
province_summary <- summarise_points_by_polygon(
polygons = nl_provincie,
points = insurance,
value = "amount",
fun = sum,
outside = "ignore"
)
sf::st_drop_geometry(province_summary)[, c("areaname", "amount_sum")]
#> areaname amount_sum
#> 1 Drenthe 56766689
#> 2 Flevoland 55795037
#> 3 Friesland 78581984
#> 4 Gelderland 269468412
#> 5 Groningen 106580080
#> 6 Limburg 140680821
#> 7 Noord-Brabant 377776132
#> 8 Noord-Holland 593255924
#> 9 Overijssel 148939513
#> 10 Utrecht 226377123
#> 11 Zeeland 82251913
#> 12 Zuid-Holland 697040028This polygon summary answers a different question from the hotspot search. The hotspot search is based on circles with fixed radius; the polygon summary is based on predefined administrative boundaries. Both can be useful, but they should not be interpreted as the same measure.
The radius should be chosen from the application context. In applied concentration analysis, it may follow from a scenario definition, reporting objective, risk appetite definition, hazard radius, service radius, or another domain-specific choice. The package computes the requested spatial aggregates; the analyst determines which exposure measure, radius, portfolio scope, and assumptions are appropriate for the application.
The coordinate columns supplied to the functions are assumed to be
longitude and latitude in EPSG:4326 unless specified otherwise. Distance
calculations for hotspot optimisation are performed in a projected
coordinate reference system with metre units. The default
crs_metric = 3035 is suitable for Europe-wide applications;
for other regions, use a metric CRS appropriate to the study area.
For the continuous and grid methods,
cell_size controls the initial screening resolution.
Smaller cells give a finer screening surface but increase computation
time. The value should be positive and no larger than the radius. For
radius = 200, a cell_size such as 100 metres
is a practical starting point.
For large portfolios, it is useful to keep a reproducible record of:
The concentration hotspot problem in spatialrisk can be
interpreted as a fixed-radius circle-placement problem. Given a set of
insured locations, such as buildings or other point-represented risks,
each location has an associated value, for example insured amount,
exposure, premium, or loss. The objective is to find the location of a
circle with fixed radius that maximizes the total value of the points
contained in that circle.
This problem is closely related to the circle placement problem studied by Chazelle and Lee (1986). In their formulation, a set of weighted points in the plane is given and a disk of fixed radius must be placed such that the total covered weight is maximized. This provides the theoretical basis for using boundary and pair-intersection geometry.
For the underlying continuous fixed-radius problem, an optimum can be
represented by a centre associated with the relevant point and
circle-boundary intersection geometry under the usual assumptions:
observations are points, weights are non-negative, distances are
Euclidean in a projected coordinate reference system, and the radius is
fixed. The candidate set consisting of observed point locations and the
intersections of radius-r circles around pairs of
observations is sufficient for the single-disk optimum under those
assumptions.
The practical continuous implementation uses spatial
screening and local refinement to avoid evaluating the full candidate
set indiscriminately. With non-negative weights, the default
automatically calculated lower bound, complete pair-intersection
refinement, and exact scoring without grid fallback, the screening step
is optimality-preserving for the single-disk problem. The expanded focal
value upper-bounds every centre in its cell, while the automatic
threshold is the value of an actually feasible centre; a cell below that
threshold therefore cannot contain a strictly better solution. All
observed and pair-intersection candidates whose own cells survive are
represented in the exact sweep and evaluated against the complete active
portfolio. Under these conditions, cell_size affects
screening resolution and computational cost, but not the global-optimum
guarantee.
The lower-level call
optimize_hotspot(prepare_spatialrisk(...)) provides that
complete geometric candidate search for the active portfolio. For the
first hotspot, under the point, non-negative-weight, fixed-radius, and
projected-Euclidean assumptions stated above, this is the full finite
candidate characterisation of the one-disk problem. The same global
guarantee applies to the screened workflow only under the conditions
stated above. It does not apply to a user-supplied threshold, negative
weights, grid fallback, grid search, or the joint placement of multiple
circles.
For insurance applications this is useful because the method directly targets accumulation risk: the maximum total value that can be found within a specified distance of any location. This may be used, for example, to identify local concentrations of insured building values, exposed sums insured, or other portfolio-level risk measures.
For multiple hotspots, spatialrisk follows a greedy
approach: after the first hotspot is selected, its covered points are
removed and the next hotspot is computed on the remaining portfolio.
Each step solves the corresponding single-hotspot search on the
remaining data according to the selected method and settings, but the
sequence is not necessarily globally optimal as a joint multi-circle
optimisation problem.
Chazelle, B. M. and Lee, D. T. (1986). On a circle placement problem. Computing, 36(1–2), 1–16. doi:10.1007/BF02238188.