# Fixed-radius concentration analysis

## Motivation

Insurance portfolios often contain many point-level exposures, such as buildings or policies with an insured amount. For concentration risk management, the relevant question is not only the total portfolio value, but also how much value can accumulate locally.

A common applied task is therefore to determine the largest total value within a circle of fixed radius. This type of calculation is useful when internal risk limits or regulatory requirements define concentration in terms of exposure within a specified distance, for example a 200 metre radius.

The purpose of `spatialrisk` is to make this workflow reproducible in R:

- inspect which observations fall within a radius;
- calculate fixed-radius sums around selected target locations;
- identify concentration hotspots;
- summarise point-level values to polygons for reporting.

The package does not impose a probabilistic model. It computes deterministic spatial aggregates from observed point locations and values.

## Example portfolio

The examples below use the included `Groningen` data. For speed in this vignette, we use a small subset. 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.

## Inspect observations within a radius

Before searching for a maximum, it is useful to inspect the local aggregation rule. The following call identifies all points within 200 metres of a chosen centre.

    local_points <- points_within_radius(
      portfolio,
      lon_center = 6.5549,
      lat_center = 53.1942,
      radius = 200
    )

    head(local_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(local_points)
    #> [1] 110
    sum(local_points$amount)
    #> [1] 25668

The returned data contains the observations that contribute to this local fixed-radius sum. This is helpful for auditability: the aggregate can be traced back to the underlying policies or locations.

## Calculate fixed-radius sums around target 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 grid points, or at any other candidate centres.

## Identify a concentration hotspot

The main applied task is to find the location where the fixed-radius sum is largest. `concentration_hotspot()` searches for such a centre and returns both the hotspot and the contributing observations. The default `method = "continuous"` searches for a centre that may lie between buildings. Internally, it uses a coarse spatial screening step followed by local pair-intersection refinement. If the local refinement subset is larger than `max_refinement_points`, the function falls back to grid refinement.

    hotspot <- concentration_hotspot(
      portfolio,
      value = "amount",
      radius = 200,
      cell_size = 100,
      progress = FALSE,
      top_n = 2
    )

    plot(hotspot)

The reported `amount_sum` is the sum of `amount` within the 200 metre circle around the selected centre. The argument `top_n` gives the number of hotspots to return. When `top_n > 1`, the points contributing to the first hotspot are removed before the next hotspot is searched for. This gives non-overlapping hotspot assignments. The contributing observations are available as follows:

    head(hotspot$contributing_points[, c("id", "data_row", "lon", "lat", "amount", "amount_sum")])
    #> # A tibble: 6 × 6
    #>      id data_row   lon   lat amount amount_sum
    #>   <int>    <int> <dbl> <dbl>  <dbl>      <dbl>
    #> 1     1     1492  6.55  53.2    148      64308
    #> 2     1     4703  6.55  53.2    132      64308
    #> 3     1    18287  6.55  53.2    130      64308
    #> 4     1    19958  6.55  53.2    138      64308
    #> 5     1    22587  6.55  53.2    142      64308
    #> 6     1       19  6.54  53.2    411      64308

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 default continuous method may place the hotspot centre between buildings. This is important: the circle with the largest total value often does not have its centre exactly on one observed building, but somewhere between several buildings.

The package also includes an observed-points method, available by setting `method = "observed"` in `concentration_hotspot()`. This method searches only observed point locations as possible circle centres. It is useful as a fast and deterministic benchmark, but it can 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 6.547323 53.23663      64308
    #> observed    1 6.547288 53.23664      64172

In this example the continuous hotspot has a higher `amount_sum` than the observed-points hotspot, because the observed method only evaluates existing building locations as candidate centres.

    plot(hotspot_continuous)

    plot(hotspot_observed)

The original grid-refinement workflow remains available with `method = "grid"`. In that method, `grid_precision` controls the local grid refinement. For the default `method = "continuous"`, `grid_precision` is only used if the local pair-refinement subset is too large and the function falls back to grid refinement.

## Reporting by polygon

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 insurance concentration analysis, it may follow from regulation, internal risk appetite, or a scenario definition.

The coordinate columns supplied to the functions are assumed to be longitude and latitude in EPSG:4326 unless specified otherwise. Distance calculations for point-level radius operations are performed in metres.

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.

## Relation to location-allocation problems

The fixed-radius hotspot problem is related to the maximum covering location problem. In the present package, the emphasis is applied: calculating and auditing concentration values for spatial portfolios. The implementation is intended to support this workflow rather than provide a general optimisation framework.