# Distance-based Ordination Methods

### DISTANCE-BASED ORDINATION METHODS

Distance-based approaches rely on a square, symmetric distance matrix or similarity matrix. See Similarity, Distance and Difference. For polar ordination, it is necessary for data to obey the triangle inequality (i.e. the distance between A and B plus the distance between B and C cannot exceed the distance between A and C). Unlike methods derived from eigenanalysis, distance-based methods do not provide species and sample scores simultaneously.

## Polar ordination (PO)

• Also known as Bray-Curtis Ordination or Wisconsin Ordination.
• PO is perhaps the easiest ordination technique to visualize, and it is possible to perform without a computer.
• It is possible for axes to be correlated.
• The user must pre-specify samples as end points (alternatively, there are automatic ways endpoints can be specified). These endpoints have a privileged position; all other samples are defined relative to them.
• PO is the only technique where one can control the direction of gradients (i.e. left vs. right or up vs. down).
• New samples can be added without affecting the ordination.

## Principal Coordinates Analysis (PCoA)

• This technique has been labelled "Multidimensional Scaling", although this term might be better avoided, since it is now becoming synonymous with "Nonmetric Multidimensional Scaling".  PCoA has also been termed "Metric Multidimensional Scaling".
• PCoA maximizes the linear correlation between distance measures and distance in the ordination.
• One cannot easily put new points in a PCoA.
• PCoA (and NMDS) can be useful if one has only a distance (or similarity) matrix (e.g. for DNA hybridization, or for plotless sampling)
• The underlying model for PCoA (and NMDS) is that there a fixed number of gradients. In contrast, PCA, RA, and DCA assume that there are potentially many gradients, but of declining importance.
• Special case: for Euclidean distance, PCoA = PCA.
• PCoA can be expressed as an eigenanalysis.

## Nonmetric Multidimensional Scaling (NMDS)

• This is sometimes labeled "Multidimensional Scaling (MDS)", although this term has been used for PCoA.
• NMDS is very computer intensive; it has only recently become feasible for large data sets on the microcomputer.
• NMDS Maximizes rank-order correlation between distance measures and distance in ordination space. Points are moved to minimize "stress". Stress is a measure of the mismatch between the two kinds of distance.
• NOTE: Gauch (1982) is WRONG is stating that NMDS assumes species have monotonic relationships to gradients. What it assumes is that dissimilarity is monotonically related to ecological distance.
• The user must pre-specify number of dimensions, or examine a plot of "stress" as a function of number of axes, and select the number of dimensions a posteriori.
• The configuration will change as the number of axes change.
• There is no guarantee that the correct (lowest stress) solution will be found, though it is widely assumed that this is not a big problem.
• To increase the likelihood of finding the correct solution, a DCA is often performed first.
• A recent variant of NMDS is Local NMDS, which only minimizes stress locally.

## Eigenanalysis-based methods

• It can be shown that some methods described in eigenanalysis-based ordination methods are also special cases of distance-based methods, where the distance is based on the chi-squared distribution. While this is true, it is worth keeping in mind that the underlying philosophy of the eigenanalysis-based methods is fundamentally different: these methods attempt to faithfully place species along gradients (either inferred or directly related to measured variables), and not to faithfully relate difference to distance.

## Polar Ordination

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