# Principal Coordinates of Neighbourhood Matrix

### Description

This function computed classical PCNM by the principal coordinate analysis of a truncated distance matrix. These are commonly used to transform (spatial) distances to rectangular data that suitable for constrained ordination or regression.

### Usage

1 |

### Arguments

`dis` |
A distance matrix. |

`threshold` |
A threshold value or truncation distance. If
missing, minimum distance giving connected network will be
used. This is found as the longest distance in the minimum spanning
tree of |

`w` |
Prior weights for rows. |

`dist.ret` |
Return the distances used to calculate the PCNMs. |

### Details

Principal Coordinates of Neighbourhood Matrix (PCNM) map distances between rows onto rectangular matrix on rows using a truncation threshold for long distances (Borcard & Legendre 2002). If original distances were Euclidean distances in two dimensions (like normal spatial distances), they could be mapped onto two dimensions if there is no truncation of distances. Because of truncation, there will be a higher number of principal coordinates. The selection of truncation distance has a huge influence on the PCNM vectors. The default is to use the longest distance to keep data connected. The distances above truncation threshold are given an arbitrary value of 4 times threshold. For regular data, the first PCNM vectors show a wide scale variation and later PCNM vectors show smaller scale variation (Borcard & Legendre 2002), but for irregular data the interpretation is not as clear.

The PCNM functions are used to express distances in rectangular form
that is similar to normal explanatory variables used in, e.g.,
constrained ordination (`rda`

, `cca`

and
`capscale`

) or univariate regression (`lm`

)
together with environmental variables (row weights should be supplied
with `cca`

; see Examples). This is regarded as a more
powerful method than forcing rectangular environmental data into
distances and using them in partial mantel analysis
(`mantel.partial`

) together with geographic distances
(Legendre et al. 2008, but see Tuomisto & Ruokolainen 2008).

The function is based on `pcnm`

function in Dray's unreleased
spacemakeR package. The differences are that the current
function uses `spantree`

as an internal support
function. The current function also can use prior weights for rows by
using weighted metric scaling of `wcmdscale`

. The use of
row weights allows finding orthonormal PCNMs also for correspondence
analysis (e.g., `cca`

).

### Value

A list of the following elements:

`values ` |
Eigenvalues obtained by the principal coordinates analysis. |

`vectors ` |
Eigenvectors obtained by the principal coordinates
analysis. They are scaled to unit norm. The vectors can be extracted
with |

`threshold` |
Truncation distance. |

`dist` |
The distance matrix where values above |

### Author(s)

Jari Oksanen, based on the code of Stephane Dray.

### References

Borcard D. and Legendre P. (2002) All-scale spatial analysis of
ecological data by means of principal coordinates of neighbour
matrices. *Ecological Modelling* **153**, 51–68.

Legendre, P., Bordard, D and Peres-Neto, P. (2008) Analyzing or
explaining beta diversity? Comment. *Ecology* **89**,
3238–3244.

Tuomisto, H. & Ruokolainen, K. (2008) Analyzing or explaining beta
diversity? A reply. *Ecology* **89**, 3244–3256.

### See Also

`spantree`

.

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 | ```
## Example from Borcard & Legendre (2002)
data(mite.xy)
pcnm1 <- pcnm(dist(mite.xy))
op <- par(mfrow=c(1,3))
## Map of PCNMs in the sample plot
ordisurf(mite.xy, scores(pcnm1, choi=1), bubble = 4, main = "PCNM 1")
ordisurf(mite.xy, scores(pcnm1, choi=2), bubble = 4, main = "PCNM 2")
ordisurf(mite.xy, scores(pcnm1, choi=3), bubble = 4, main = "PCNM 3")
par(op)
## Plot first PCNMs against each other
ordisplom(pcnm1, choices=1:4)
## Weighted PCNM for CCA
data(mite)
rs <- rowSums(mite)/sum(mite)
pcnmw <- pcnm(dist(mite.xy), w = rs)
ord <- cca(mite ~ scores(pcnmw))
## Multiscale ordination: residual variance should have no distance
## trend
msoplot(mso(ord, mite.xy))
``` |

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