Fixed-radius concentration analysis • spatialrisk

Motivation

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.

Example portfolio

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     28

The 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.

Quick example: find the largest concentration

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 6.547331 53.23659      64438

The 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.

plot(hotspot)

Understanding the result

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 6.547331 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      64438

This 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.

Inspecting contributing locations

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$contributing_points)
#>   id data_row      lon      lat amount distance_m amount_sum
#> 1  1     1492 6.545297 53.23569    148   168.9837      64438
#> 2  1     4703 6.545482 53.23547    132   175.7036      64438
#> 3  1    18287 6.545429 53.23546    130   178.8321      64438
#> 4  1    19958 6.545392 53.23543    138   183.0865      64438
#> 5  1    22587 6.545493 53.23545    142   176.7874      64438
#> 6  1       19 6.544724 53.23646    411   174.6183      64438
nrow(hotspot$contributing_points)
#> [1] 208
sum(hotspot$contributing_points$amount)
#> [1] 64438

The 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] 25668

The 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.

Calculating sums at specified 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        8633

The 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.

Continuous vs observed centres

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 6.547331 53.23659      64438
#> observed    1 6.547288 53.23664      64172

In 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.

plot(hotspot_continuous)

plot(hotspot_observed)

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.

Multiple hotspots

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 6.523411 53.23094      57977

The 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.

plot(hotspot_top2)

How the search works

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:

coarse spatial screening identifies promising regions;
candidate regions are retained using an automatically estimated lower bound;
candidate centres are generated or refined within those regions;
exact radius sums are evaluated for candidate centres and the best result is returned.

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 6.523411 53.23094      57977

For 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 production workflow 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 production 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       1315

The 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 only the focal candidate cells above the selected lower bound.

prepared <- prepare_spatialrisk(portfolio, value = "amount", radius = 200,
                                cell_size = 100)
plot(prepared)


selected <- select_candidates(prepared, progress = FALSE)
plot(selected)

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 first takes the highest cells from the focal raster. For those cells it runs a small local refinement step and uses the best refined concentration as the lower bound. Candidate cells are then all focal cells whose moving-window sum is at least this lower bound. 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. Within each hotspot iteration, exact candidate evaluations share one spatial lookup over the active portfolio; exact distances are calculated only for points from potentially intersecting cells. 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 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 receives the more expensive exact radius evaluation. This is more selective than retaining both centres merely because one of them lies in a candidate cell. Every retained centre is then evaluated against the full remaining active portfolio, not only against 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.

Polygon reporting

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  697040028

This 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.

Practical considerations

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 input portfolio and value column;
the radius;
the coordinate reference assumptions;
the search parameters used for hotspot detection;
the observations contributing to the reported hotspot.

Mathematical background

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 first single-circle optimum under those assumptions.

The practical continuous implementation uses spatial screening and local refinement to avoid evaluating the full candidate set indiscriminately. Computational settings such as cell_size, the candidate lower bound, and max_refinement_points determine how extensively the candidate space is explored. The pair-intersection refinement is exact within the screened local candidate areas; it is not the same as evaluating every possible pair-intersection candidate globally in every call.

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. This statement does not apply to the screened workflow, grid fallback, 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.

Reference

Chazelle, B. M. and Lee, D. T. (1986). On a circle placement problem. Computing, 36(1–2), 1–16. doi:10.1007/BF02238188.