Displays the Fry plot (Patterson plot) of a spatial point pattern.

```
fryplot(X, ..., width=NULL, from=NULL, to=NULL, axes=FALSE)
frypoints(X, from=NULL, to=NULL, dmax=Inf)
```

`fryplot`

returns `NULL`

.

`frypoints`

returns a point pattern (object of class `"ppp"`

).

- X
A point pattern (object of class

`"ppp"`

) or something acceptable to`as.ppp`

.- ...
Optional arguments to control the appearance of the plot.

- width
Optional parameter indicating the width of a box for a zoomed-in view of the Fry plot near the origin.

- from,to
Optional. Subset indices specifying which points of

`X`

will be considered when forming the vectors (drawn from each point of`from`

, to each point of`to`

.)- axes
Logical value indicating whether to draw axes, crossing at the origin.

- dmax
Maximum distance between points. Pairs at greater distances do not contribute to the result. The default means there is no maximum distance.

Adrian Baddeley Adrian.Baddeley@curtin.edu.au

and Rolf Turner r.turner@auckland.ac.nz

The function `fryplot`

generates a Fry plot (or Patterson plot);
`frypoints`

returns the points of the Fry plot as a point pattern
dataset.

Fry (1979) and Hanna and Fry (1979) introduced a manual graphical method for
investigating features of a spatial point pattern of mineral deposits.
A transparent sheet, marked
with an origin or centre point, is placed over the point pattern.
The transparent sheet is shifted so that the origin lies over one of
the data points, and the positions of all the *other* data points
are copied onto the transparent sheet. This procedure is repeated for
each data point in turn. The resulting plot (the Fry plot)
is a pattern of \(n(n-1)\) points, where \(n\) is the original number
of data points. This procedure was previously proposed by
Patterson (1934, 1935) for studying inter-atomic distances in
crystals, and is also known as a Patterson plot.

The function `fryplot`

generates the Fry/Patterson plot.
Standard graphical parameters
such as `main`

, `pch`

,
`lwd`

, `col`

, `bg`

, `cex`

can be used to control
the appearance of the plot.
To zoom in (to view only a subset of the Fry plot at higher
magnification), use the argument `width`

to specify the width
of a rectangular field of view centred at the origin, or the standard
graphical arguments `xlim`

and `ylim`

to specify another
rectangular field of view. (The actual field of view may be slightly
larger, depending on the graphics device.)

The function `frypoints`

returns the points of the Fry
plot as a point pattern object. There may be a large number of points
in this pattern, so this function should be used only if further
analysis of the Fry plot is required.

Fry plots are particularly useful for recognising anisotropy in regular point patterns. A void around the origin in the Fry plot suggests regularity (inhibition between points) and the shape of the void gives a clue to anisotropy in the pattern. Fry plots are also useful for detecting periodicity or rounding of the spatial coordinates.

In mathematical terms, the Fry plot of a point pattern `X`

is simply a plot of the vectors `X[i] - X[j]`

connecting all
pairs of distinct points in `X`

.

The Fry plot is related to the \(K\) function (see
`Kest`

) and the reduced second moment measure
(see `Kmeasure`

). For example, the number
of points in the Fry plot lying within a circle of given radius
is an unnormalised and uncorrected version of the \(K\) function.
The Fry plot has a similar appearance to the plot of the
reduced second moment measure `Kmeasure`

when the
smoothing parameter `sigma`

is very small.

The Fry plot does not adjust for the effect
of the size and shape of the sampling window.
The density of points in the Fry plot tapers off near the edges of the
plot. This is an edge effect, a consequence of the bounded sampling
window. In geological applications this is usually not
important, because interest is focused on the behaviour near the
origin where edge effects can be ignored.
To correct for the edge effect, use `Kmeasure`

or
`Kest`

or its relatives.

Fry, N. (1979)
Random point distributions and strain measurement in rocks.
*Tectonophysics* **60**, 89--105.

Hanna, S.S. and Fry, N. (1979)
A comparison of methods of strain determination in rocks from
southwest Dyfed (Pembrokeshire) and adjacent areas.
*Journal of Structural Geology* **1**, 155--162.

Patterson, A.L. (1934) A Fourier series method for the determination
of the component of inter-atomic distances in crystals.
*Physics Reviews* **46**, 372--376.

Patterson, A.L. (1935) A direct method for the determination of the
components of inter-atomic distances in crystals.
*Zeitschrift fuer Krystallographie* **90**, 517--554.

`Kmeasure`

,
`Kest`

```
## unmarked data
fryplot(cells)
Y <- frypoints(cells)
## numerical marks
fryplot(longleaf, width=4, axes=TRUE)
## multitype points
fryplot(amacrine, width=0.2,
from=(marks(amacrine) == "on"),
chars=c(3,16), cols=2:3,
main="Fry plot centred at an On-cell")
points(0,0)
```

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