---
title: "Fixed-radius concentration analysis"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Fixed-radius concentration analysis}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include = FALSE}
knitr::opts_chunk$set(
  fig.path = "figures/fixed-radius-concentration-",
  collapse = TRUE,
  comment = "#>"
)
```

## 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.

```{r}
library(spatialrisk)

portfolio <- Groningen
portfolio <- portfolio[, c("lon", "lat", "amount")]

head(portfolio)
```

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.

```{r}
local_points <- points_within_radius(
  portfolio,
  lon_center = 6.5549,
  lat_center = 53.1942,
  radius = 200
)

head(local_points)
nrow(local_points)
sum(local_points$amount)
```

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.

```{r}

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
```

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.

```{r}

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:

```{r}
head(hotspot$contributing_points[, c("id", "data_row", "lon", "lat", "amount", "amount_sum")])
```

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.

```{r}
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
)

```

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.

```{r hotspot-method-comparison, eval = requireNamespace("mapview", quietly = TRUE)}

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.

```{r, message = FALSE}
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")]
```

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.