Methods Accepting All Measurement Levels

GOWER

Gower’s similarity

is computed as follows:

For nominal, ordinal, interval, or ratio variable,

For asymmetric nominal variable,

For nominal or asymmetric nominal variable,

For ordinal, interval, or ratio variable,

DGOWER

1 minus Gower

Methods Accepting Ratio, Interval, and Ordinal Variables

EUCLID

Euclidean distance

SQEUCLID

squared Euclidean distance

SIZE

size distance

SHAPE

shape distance

Note:squared shape distance plus squared size distance equals squared Euclidean distance.

COV

covariance similarity coefficient
, where

CORR

correlation similarity coefficient

DCORR

correlation transformed to Euclidean distance as sqrt(1–CORR)

SQCORR

squared correlation

DSQCORR

squared correlation transformed to squared Euclidean distance as (1–SQCORR)

L()

Minkowski () distance, where is a positive numeric value

CITYBLOCK

CHEBYCHEV

POWER()

generalized Euclidean distance, where is a nonnegative numeric value and is a positive numeric value. The distance between two observations is the th root of sum of the absolute differences to the th power between the values for the observations:

Methods Accepting Ratio Variables

SIMRATIO

similarity ratio

DISRATIO

one minus similarity ratio

NONMETRIC

Lance-Williams nonmetric coefficient

CANBERRA

Canberra metric coefficient. See Sneath and Sokal (1973), pp. 125–126.

COSINE

cosine coefficient

DOT

dot (inner) product coefficient

OVERLAP

sum of the minimum values

DOVERLAP

maximum of the sum of the and the sum of minus overlap

CHISQ

chi-squared
If the data represent the frequency counts, chi-squared dissimilarity between two sets of frequencies can be computed. A 2-by- contingency table is illustrated to explain how the chi-squared dissimilarity is computed as follows:

	Variable				Row
Observation	Var 1	Var 2	...	Var v	Sum
X			...
Y			...
Column Sum			...

where

The chi-squared measure is computed as follows:

where for = 1, 2, ...,

CHI

squared root of chi-squared

PHISQ

phi-squared
This is the CHISQ dissimilarity normalized by the sum of weights

PHI

squared root of phi-squared

Methods Accepting Symmetric Nominal Variables

The following notation is used for computing to . Notice that only the nonmissing pairs are discussed below; all the pairs with at least one missing value will be excluded from any of the computations in the following section because

nonmissing matches

, where

nonmissing mismatches

, where

total nonmissing pairs

HAMMING

Hamming distance

MATCH

simple matching coefficient

DMATCH

simple matching coefficient transformed to Euclidean distance

DSQMATCH

simple matching coefficient transformed to squared Euclidean distanc

HAMANN

Hamann coefficient

RT

Roger and Tanimoto

SS1

Sokal and Sneath 1

SS3

Sokal and Sneath 3. The coefficient between an observation and itself is always indeterminate (missing) since there is no mismatch.

The following notation is used for computing to . Notice that only the nonmissing pairs are discussed in the following section; all the pairs with at least one missing value are excluded from any of the computations in the following section because

Also, the observed nonmissing data of an asymmetric binary variable can have only two possible outcomes: presence or absence. Therefore, the notation, PX (present mismatches), always has a value of zero for an asymmetric binary variable.

The following methods distinguish between the presence and absence of attributes.

mismatches with at least one present

, where

present matches

, where

present mismatches

, where

both present =

at least one present =

present-absent mismatches

, where

total nonmissing pairs

Methods Accepting Asymmetric Nominal and Ratio Variables

JACCARD

Jaccard similarity coefficient

The JACCARD method is equivalent to the SIMRATIO method if there are only ratio variables; if there are both ratio and asymmetric nominal variables, the coefficient is computed as sum of the coefficient from the ratio variables (SIMRATIO) and the coefficient from the asymmetric nominal variables.

DJACCARD

Jaccard dissimilarity coefficient

The DJACCARD method is equivalent to the DISRATIO method if there are only ratio variables; if there are both ratio and asymmetric nominal variables, the coefficient is computed as sum of the coefficient from the ratio variables (DISRATIO) and the coefficient from the asymmetric nominal variables.

Methods Accepting Asymmetric Nominal Variables

DICE

Dice coefficient or Czekanowski/Sorensen similarity coefficient

RR

Russell and Rao. This is the binary equivalent of the dot product coefficient.

BLWNM

BRAYCURTIS

Binary Lance and Williams, also known as Bray and Curtis coefficient

K1

Kulcynski 1. The coefficient between an observation and itself is always indeterminate (missing) since there is no mismatch.