Description Usage Arguments Details Value Author(s) References See Also Examples

Simulates point patterns according to the null hypothesis and returns the envelope of *Kd* according to the confidence level.

1 2 3 4 5 |

`X` |
A point pattern ( |

`r` |
A vector of distances. If |

`NumberOfSimulations` |
The number of simulations to run, 100 by default. |

`Alpha` |
The risk level, 5% by default. |

`ReferenceType` |
One of the point types. |

`NeighborType` |
One of the point types. By default, the same as reference type. |

`Weighted` |
Logical; if |

`Original` |
Logical; if |

`Approximate` |
if not 0 (1 is a good choice), exact distances between pairs of points are rounded to 1024 times |

`Adjust` |
Force the automatically selected bandwidth (following Silverman, 1986) to be multiplied by |

`MaxRange` |
The maximum value of |

`StartFromMinR` |
Logical; if |

`SimulationType` |
A string describing the null hypothesis to simulate. The null hypothesis may be
" |

`Global` |
Logical; if |

This envelope is local by default, that is to say it is computed separately at each distance. See Loosmore and Ford (2006) for a discussion.

The global envelope is calculated by iteration: the simulations reaching one of the upper or lower values at any distance are eliminated at each step. The process is repeated until *Alpha / Number of simulations* simulations are dropped. The remaining upper and lower bounds at all distances constitute the global envelope. Interpolation is used if the exact ratio cannot be reached.

An envelope object (`envelope`

). There are methods for print and plot for this class.

The `fv`

contains the observed value of the function, its average simulated value and the confidence envelope.

Eric Marcon <[email protected]>

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

Kenkel, N. C. (1988). Pattern of Self-Thinning in Jack Pine: Testing the Random Mortality Hypothesis. *Ecology* 69(4): 1017-1024.

Loosmore, N. B. and Ford, E. D. (2006). Statistical inference using the G or K point pattern spatial statistics. *Ecology* 87(8): 1925-1931.

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.

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

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```
Loading required package: spatstat
Loading required package: nlme
Loading required package: rpart
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.
Loading required package: Rcpp
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.
Generating 100 simulations by evaluating expression ...
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100.
Done.
Warning message:
system call failed: Cannot allocate memory
```

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