Euclidean distances

Euclidean distance, an appropriate measure for environmental (and other) data types, is defined as:

$$ D_1 = \sqrt{ \sum_i \left( y _ {i1} - y_{i2} \right) ^ 2 } $$

where the $y_{i1}$ & $y_{i2}$ result from pre-treatment by transformation (sometimes) and subsequent normalisation (often). The outcome is a triangular distance matrix, which orders in the opposite direction to similarity: high similarity = low distance (= low dissimilarity). Note, however, that the user does not have to worry about which way round the resemblances are ordered: all routines will utilise the information given in the Resemblance type to make sensible choices.

Re-open the Ekofisk workspace Ekofisk ws from the \Ekofisk macrofauna directory; you should have available the transformed and normalised environmental data (Data4 perhaps) from Section 4, on which to calculate Euclidean distance. The Analyse>Resemblance dialog box now gives the default as Measure•Euclidean because Data type has been defined as Environmental, so you can take the defaults here. The result is a resemblance matrix of type Distance; the History box on the Edit>Properties dialog shows its derivation as Euclidean distance on normalised data. Compute Manhattan distance also (see next page) and rename the sheets as Euclid and Manhattan by clicking (twice, slowly) on their default Resem names in the Explorer tree. Most other measures in the lists below are not suitable for normalised environmental data but are designed for positive ‘quantities’.