# Kdhat: Estimation of the Kd function In dbmss: Distance-Based Measures of Spatial Structures

## Description

Estimates the Kd function

## Usage

 ```1 2 3``` ``` Kdhat(X, r = NULL, ReferenceType, NeighborType = ReferenceType, Weighted = FALSE, Original = TRUE, Approximate = ifelse(X\$n < 10000, 0, 1), Adjust = 1, MaxRange = "ThirdW", StartFromMinR = FALSE, CheckArguments = TRUE) ```

## Arguments

 `X` A weighted, marked planar point pattern (`wmppp.object`) or a `Dtable` object. `r` A vector of distances. If `NULL`, a default value is set: 512 equally spaced values are used, from the smallest distance between points to half the diameter of the window. `ReferenceType` One of the point types. If "", all points are considered (this is not the default value; `NeighborType` is ignored then) to estimate the average value of simulated Kd values under the null hypothesis of RandomLocation (Marcon and Puech, 2012). `NeighborType` One of the point types. By default, the same as reference type. `Weighted` Logical; if `TRUE`, estimates the Kemp function. `Original` Logical; if `TRUE` (by default), the original bandwidth selection by Duranton and Overman (2005) following Silverman (1986: eq 3.31) is used. If `FALSE`, it is calculated following Sheather and Jones (1991), i.e. the state of the art. See `bw.SJ` for more details. `Approximate` if not 0 (1 is a good choice), exact distances between pairs of points are rounded to 1024 times `Approximate` single values equally spaced between 0 and the largest distance. This technique (Scholl and Brenner, 2015) allows saving a lot of memory when addressing large point sets (the default value is 1 over 10000 points). Increasing `Approximate` allows better precision at the cost of proportional memory use. Ignored if `X` is a `Dtable` object. `Adjust` Force the automatically selected bandwidth (following `Original`) to be multiplied by `Adjust`. Setting it to values lower than one (1/2 for example) will sharpen the estimation. `MaxRange` The maximum value of `r` to consider, ignored if `r` is not `NULL`. Default is "ThirdW", one third of the diameter of the window. Other choices are "HalfW", and "QuarterW" and "D02005". "HalfW", and "QuarterW" are for half or the quarter of the diameter of the window. "D02005" is for the median distance observed between points, following Duranton and Overman (2005). "ThirdW" should be close to "DO2005" but has the advantage to be independent of the point types chosen as `ReferenceType` and `NeighborType`, to simplify comparisons between different types. "D02005" is approximated by "ThirdW" if `Approximate` is not 0. if `X` is a `Dtable` object, the diameter of the window is taken as the max distance between points. `StartFromMinR` Logical; if `TRUE`, points are assumed to be further from each other than the minimum observed distance, So Kd will not be estimated below it: it is assumed to be 0. If `FALSE`, distances are smoothed down to \$r=0\$. Ignored if `Approximate` is not 0: then, estimation always starts from \$r=0\$. `CheckArguments` Logical; if `TRUE`, the function arguments are verified. Should be set to `FALSE` to save time in simulations for example, when the arguments have been checked elsewhere.

## Details

Kd is a density, absolute measure of a point pattern structure. Kd is computed efficiently by building a matrix of distances between point pairs and calculating the density of their distribution (the default values of `r` are those of the `density` function). The kernel estimator is Gaussian.

The weighted Kd function has been named Kemp (emp is for employees) by Duranton and Overman (2005).

If `X` is not a `Dtable` object, the maximum value of `r` is obtained from the geometry of the window rather than caculating the median distance between points as suggested by Duranton and Overman (2005) to save (a lot of) calculation time.

## Value

An object of class `fv`, see `fv.object`, which can be plotted directly using `plot.fv`.

## Note

Estimating Kd relies on calculating distances, exactly or approximately (if `Approximate` is not 0). Then distances are smoothed by estimating their probability density. Reflection is used to estimate density close to the lowest distance, that is the minimum observed distance (if `StartFromMinR` is `TRUE`) or 0: all distances below 4 times the estimation kernel bandwith apart from the lowest distance are duplicated (symmetrically with respect to the lowest distance) to avoid edge effects (underestimation of the density close to the lowest distance).

## Author(s)

Eric Marcon <[email protected]>

## References

Duranton, G. and Overman, H. G. (2005). Testing for Localisation Using Micro-Geographic Data. Review of Economic Studies 72(4): 1077-1106.

Marcon, E. and F. Puech (2017). A typology of distance-based measures of spatial concentration. Regional Science and Urban Economics. 62:56-67.

Scholl, T. and Brenner, T. (2015) Optimizing distance-based methods for large data sets, Journal of Geographical Systems 17(4): 333-351.

Sheather, S. J. and Jones, M. C. (1991) A reliable data-based bandwidth selection method for kernel density estimation. Journal of the Royal Statistical Society series B, 53, 683-690.

Silverman, B. W. (1986). Density estimation for statistics and data analysis. Chapman and Hall, London.

`KdEnvelope`, `Mhat`

## Examples

 ```1 2 3 4 5 6 7``` ```data(paracou16) plot(paracou16) # Calculate Kd (Paracou <- Kdhat(paracou16, , "Q. Rosea", "V. Americana")) # Plot plot(Paracou) ```

### Example output

```Loading required package: spatstat

spatstat 1.52-1       (nickname: 'Apophenia')
For an introduction to spatstat, type 'beginner'

Note: spatstat version 1.52-1 is out of date by more than 7 months; we recommend upgrading to the latest version.
Warning message:
Interpretation of arguments maxsize and markscale has changed (in spatstat version 1.37-0 and later). Size of a circle is now measured by its diameter.
Function value object (class 'fv')
for the function r -> Kd(r)
.................................
Math.label Description
r  r          distance argument r
Kd hat(Kd)(r) Estimated Kd(r)
.................................
Default plot formula:  .~r
where "." stands for 'Kd'
Recommended range of argument r: [0, 102.11]
Available range of argument r: [0, 102.11]
Unit of length: 1 meter
```

dbmss documentation built on March 19, 2018, 5:05 p.m.