# Tstat

##### Third order summary statistic

Computes the third order summary statistic $T(r)$ of a spatial point pattern.

- Keywords
- spatial, nonparametric

##### Usage

```
Tstat(X, ..., r = NULL, rmax = NULL,
correction = c("border", "translate"), ratio = FALSE, verbose=TRUE)
```

##### Arguments

- X
- The observed point pattern,
from which an estimate of $T(r)$ will be computed.
An object of class
`"ppp"`

, or data in any format acceptable to`as.ppp()`

. - ...
- Ignored.
- r
- Optional. Vector of values for the argument $r$ at which $T(r)$
should be evaluated. Users are advised
*not*to specify this argument; there is a sensible default. - rmax
- Optional. Numeric. The maximum value of $r$ for which $T(r)$ should be estimated.
- correction
- Optional. A character vector containing any selection of the
options
`"none"`

,`"border"`

,`"bord.modif"`

,`"translate"`

, or`"best"`

. It specifies the edge correction(s) to be applied. - ratio
- Logical.
If
`TRUE`

, the numerator and denominator of each edge-corrected estimate will also be saved, for use in analysing replicated point patterns. - verbose
- Logical. If
`TRUE`

, an estimate of the computation time is printed.

##### Details

This command calculates the third-order summary statistic $T(r)$ for a spatial point patterns, defined by Schladitz and Baddeley (2000).

The definition of $T(r)$ is similar to the definition of Ripley's $K$ function $K(r)$, except that $K(r)$ counts pairs of points while $T(r)$ counts triples of points. Essentially $T(r)$ is a rescaled cumulative distribution function of the diameters of triangles in the point pattern. The diameter of a triangle is the length of its longest side.

##### Value

##### Computation time

If the number of points is large, the algorithm can take a very long time
to inspect all possible triangles. A rough estimate
of the total computation time will be printed at the beginning
of the calculation. If this estimate seems very large,
stop the calculation using the user interrupt signal, and
call `Tstat`

again, using `rmax`

to restrict the
range of `r`

values,
thus reducing the number of triangles to be inspected.

##### References

Schladitz, K. and Baddeley, A. (2000)
A third order point process characteristic.
*Scandinavian Journal of Statistics* **27** (2000) 657--671.

##### See Also

##### Examples

`plot(Tstat(redwood))`

*Documentation reproduced from package spatstat, version 1.25-1, License: GPL (>= 2)*