SAS/STAT 9.1 Users Guide Volume 2 only

v

the number of variables or the dimensionality

x _j

data for observation x and the j th variable, where j = 1 to v

y _j

data for observation y and the j th variable, where j = 1 to v

w _j

weight for the j th variable from the WEIGHTS= option in the VAR statement. w _j = 0 when either x _j or y _j is missing.

W

the sum of total weights. No matter if the observation is missing or not, its weight is added to this metric.

x

mean for observation x

y

mean for observation y

d ( x,y )

the distance or dissimilarity between observations x and y

s ( x,y )

the similarity between observations x and y

GOWER

Gowers similarity

To compute :

for nominal, ordinal, interval, or ratio variable,

= 1

for asymmetric nominal variable,

= 1, if either x _j or y _j is present

= 0, if both x _j and y _j are absent

To compute :

for nominal or asymmetric nominal variable,

= 1, if x _j = y _j

= 0, if x _j   y _j ;

for ordinal (where data are replaced by corresponding rank scores), interval, or ratio variable,

= 1 ˆ’ x _j ˆ’ y _j

DGOWER

1 minus Gower

d ₂ ( x,y ) = 1 ˆ’ s ₁ ( x,y )

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

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(p)

Minkowski ( L _p ) distance, where p is a positive numeric value

CITYBLOCK

L ₁

CHEBYCHEV

L _ˆ

POWER( p, r )

Generalized Euclidean distance, where p is a non-negative numeric value, and r is a positive numeric value. The distance between two observations is the r th root of sum of the absolute differences to the p th power between the values for the observations

SIMRATIO

Similarity ratio

DISRATIO

one minus similarity ratio

NONMETRIC

Lance-Williams nonmetric coefficient

CANBERRA

Canberra metric coefficient

COSINE

Cosine

DOT

Dot (inner) product

OVERLAP

Sum of the minimum values

DOVERLAP

The maximum of the sum of the x and the sum of y 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 v contingency table is illustrated to explain how the chi-squared dissimilarity is computed:

Variable

Row sum

Observation

Var 1

Var 2

...

Var v

X

x ₁

x ₂

...

x _v

r _x

Y

y ₁

y ₂

...

y _v

r _y

Column sum

c ₁

c ₂

...

c _v

T

where

The chi-squared measure is computed as follows :

where for j = 1, 2, ..., v

E ( x _j )= r _x c _j /T

E ( y _j )= r _y c _j /T

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

M

non-missing matches

, where

= 1, if x _j = y _j

= 0, otherwise

X

non-missing mismatches

= 1, if x _j   y _j

= 0, otherwise

N

total non-missing pairs

HAMMING

Hamming distance

d ₂₈ ( x , y ) = X

MATCH

Simple matching coefficient

s ₂₉ ( x , y ) = M/N

DMATCH

Simple matching coefficient transformed to Euclidean distance

DSQMATCH

Simple matching coefficient transformed to squared Euclidean distance d ₃₁ ( x,y ) = 1 ˆ’ M/N = X/N

HAMANN

Hamann coefficient

s ₃₂ ( x , y ) = ( M “ X )/ N

RT

Roger and Tanimoto

s ₃₃ ( x , y ) = M /( M + 2 X )

SS1

Sokal and Sneath 1

s ₃₄ ( x , y ) = 2 M /(2 M + X )

SS3

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

s ₃₅ ( x , y ) = M / X

X

mismatches with at least one present

, where

= 1, if x _j   y _j and not both x _j and y _j are absent

= 0, otherwise

PM

present matches

, where

= 1, if x _j = y _j and both x _j and y _j are present

= 0, otherwise

PX

present mismatches

, where

= 1, if x _j   y _j and both x _j and y _j are present

= 0, otherwise

PP

both present = PM + PX

P

at least one present = PM + X

PAX

present-absent mismatches

, where

= 1, if x _j   y _j and either x _j is present and y _j is absent or x _j is absent and y _j is present

= 0 otherwise

N

total non-missing pairs

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.

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 observations and itself is always indeterminate (missing) since there is no mismatch.