The following notation is used in this section:
the number of variables or the dimensionality
data for observation x and the jth variable, where
data for observation y and the jth variable, where
weight for the jth variable from the WEIGHTS= option in the VAR statement. when either or is missing.
the sum of total weights. No matter if the observation is missing or not, its weight is added to this metric.
mean for observation x
mean for observation y
the distance or dissimilarity between observations x and y
the similarity between observations x and y
The factor is used to adjust some of the proximity measures for missing values.
Note: squared shape distance plus squared size distance equals squared Euclidean distance.
covariance similarity coefficient , where
correlation transformed to Euclidean distance as sqrt(1–CORR)
squared correlation transformed to squared Euclidean distance as (1–SQCORR)
Minkowski () distance, where p is a positive numeric value
generalized Euclidean distance, where p is a nonnegative numeric value and r is a positive numeric value. The distance between two observations is the rth root of sum of the absolute differences to the pth power between the values for the observations:
Canberra metric coefficient. See Sneath and Sokal (1973, pp. 125–126)
chisquare If the data represent the frequency counts, chisquare dissimilarity between two sets of frequencies can be computed. A 2byv contingency table is illustrated to explain how the chisquare dissimilarity is computed as follows:
Variable 
Row 

Observation 
Var 1 
Var 2 
… 
Var v 
Sum 
X 


… 


Y 


… 


Column Sum 


… 

T 
where
The chisquare measure is computed as follows:
where for j= 1, 2, …, v
phisquare This is the CHISQ dissimilarity normalized by the sum of weights
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
simple matching coefficient transformed to Euclidean distance
simple matching coefficient transformed to squared Euclidean distance
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 = PM + PX
at least one present = PM + X
presentabsent mismatches
, where
total nonmissing pairs
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.
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.
Dice coefficient or Czekanowski/Sorensen similarity coefficient
Russell and Rao. This is the binary equivalent of the dot product coefficient.
Binary Lance and Williams, also known as Bray and Curtis coefficient
Kulcynski 1. The coefficient between an observation and itself is always indeterminate (missing) since there is no mismatch.