The DISTANCE Procedure

Proximity Measures

Subsections:

The following notation is used in this section:

v

the number of variables or the dimensionality

$x_ j$

data for observation x and the jth variable, where $j = 1 \mr{\ to\ } v$

$y_ j$

data for observation y and the jth variable, where $j = 1 \mr{\ to\ } v$

$w_ j$

weight for the jth 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.

$\bar{x}$

mean for observation x

$\bar{x} = \sum _{j=1}^{v}w_ j{x_ j}/\sum _{j=1}^{v}w_ j$

$\bar{y}$

mean for observation y

$\bar{y} = \sum _{j=1}^{v}w_ j{y_ j}/\sum _{j=1}^{v}w_ j$

$d(x,y)$

the distance or dissimilarity between observations x and y

$s(x,y)$

the similarity between observations x and y

The factor $ W / \sum _{j=1}^{v}w_ j $ is used to adjust some of the proximity measures for missing values.

Methods That Accept All Measurement Levels

GOWER

Gower’s similarity $ s_{1}(x,y) = {\sum _{j=1}^{v}w_ j\delta _{x,y}^{j}d_{x,y}^ j}/ {\sum _{j=1}^{v}w_ j\delta _{x,y}^ j} $

$\delta _{x,y}^ j$ is computed as follows:

For nominal, ordinal, interval, or ratio variable,

\begin{eqnarray*} \delta _{x,y}^ j & = & 1 \end{eqnarray*}

For asymmetric nominal variable,

\begin{eqnarray*} \delta _{x,y}^ j & =& 1 \mr{,\ if\ either \ } x_ j \mr{\ or\ } y_ j \mr{\ is\ present}\\ \delta _{x,y}^ j & =& 0 \mr{,\ if\ both \ } x_ j \mr{\ and\ } y_ j \mr{\ are\ absent} \end{eqnarray*}

For nominal or asymmetric nominal variable,

\begin{eqnarray*} d_{x,y}^ j & =& 1 \mr{,\ if \ } x_ j = y_ j \\ d_{x,y}^ j & =& 0 \mr{,\ if \ } x_ j \neq y_ j \end{eqnarray*}

For ordinal, interval, or ratio variable,

\begin{eqnarray*} d_{x,y}^ j & =& 1-|x_ j-y_ j| \end{eqnarray*}
DGOWER

1 minus Gower $ d_{2}(x,y) = 1 - s_{1}(x,y) $

Methods That Accept Ratio, Interval, and Ordinal Variables

EUCLID

Euclidean distance $ d_{3}(x,y) = \sqrt { (\sum _{j=1}^ v{w_ j{(x_ j-y_ j)}^2}) W / (\sum _{j=1}^{v}w_ j)} $

SQEUCLID

squared Euclidean distance $ d_{4}(x,y) = (\sum _{j=1}^ v{w_ j{(x_ j-y_ j)}^2}) W / (\sum _{j=1}^{v}w_ j) $

SIZE

size distance $ d_{5}(x,y) = |\sum _{j=1}^ v{w_ j(x_ j-y_ j)}| \sqrt {W} / (\sum _{j=1}^{v}w_ j) $

SHAPE

shape distance $ d_{6}(x,y) = \sqrt { (\sum _{j=1}^{v}w_ j[(x_ j-\bar{x})- (y_ j-\bar{y})]^2) W / (\sum _{j=1}^{v}w_ j) }$

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

COV

covariance similarity coefficient $ s_{7}(x,y) = \sum _{j=1}^{v}w_ j(x_ j-\bar{x}) (y_ j-\bar{y})/\mi{vardiv}$, where

\begin{eqnarray*} \mi{vardiv }& =& v \mr{\ \ if \ VARDEF=N }\\ & =& v-1 \mr{\ \ if \ VARDEF=DF}\\ & =& \textstyle \sum _{j=1}^{v}w_ j \mr{\ \ if \ VARDEF=WEIGHT}\\ & =& \textstyle \sum _{j=1}^{v}w_ j-1 \mr{\ \ if \ VARDEF=WDF} \end{eqnarray*}
CORR

correlation similarity coefficient $ s_{8}(x,y) = \frac{\sum _{j=1}^{v}w_ j(x_ j-\bar{x}) (y_ j-\bar{y})}{\sqrt {\sum _{j=1}^{v}w_ j(x_ j-\bar{x})^2 \sum _{j=1}^{v}w_ j(y_ j-\bar{y})^2}} $

DCORR

correlation transformed to Euclidean distance as sqrt(1–CORR) $ d_{9}(x,y) = \sqrt {1 - s_{8}(x,y)} $

SQCORR

squared correlation $ s_{10}(x,y) = \frac{[\sum _{j=1}^{v}w_ j(x_ j-\bar{x}) (y_ j-\bar{y})]^2}{\sum _{j=1}^{v}w_ j(x_ j-\bar{x})^2 \sum _{j=1}^{v}w_ j(y_ j-\bar{y})^2} $

DSQCORR

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

$ d_{11}(x,y) = 1 - s_{10}(x,y) $

L(p)

Minkowski ($\mr{L}_ p$) distance, where p is a positive numeric value

$ d_{12}(x,y) = [ (\sum _{j=1}^ v{w_ j{|x_ j-y_ j|}^ p}) W / (\sum _{j=1}^{v}w_ j) ]^{1/p} $

CITYBLOCK

$\mr{L}_1$ $ d_{13}(x,y) = (\sum _{j=1}^ v{w_ j|x_ j-y_ j|}) W / (\sum _{j=1}^{v}w_ j) $

CHEBYCHEV

$\mr{L}_\infty $ $ d_{14}(x,y) = \max _{j=1}^{v}w_ j|x_ j-y_ j| $

POWER(p,r)

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:

$ d_{15}(x,y) = [ (\sum _{j=1}^ v{w_ j{|x_ j-y_ j| }^ p}) W / (\sum _{j=1}^{v}w_ j) ]^{1/r} $

Methods That Accept Ratio Variables

SIMRATIO

similarity ratio $ s_{16}(x,y) = \frac{\sum _ j^{v}w_ j(x_{j}y_ j)}{\sum _{j=1}^{v}w_ j(x_{j}y_ j)+\sum _ j^{v}w_ j(x_ j-y_ j)^2} $

DISRATIO

one minus similarity ratio $ d_{17}(x,y) = 1-s_{16}(x,y) $

NONMETRIC

Lance-Williams nonmetric coefficient $ d_{18}(x,y) = \frac{\sum _{j=1}^{v}w_ j|x_ j-y_ j|}{\sum _{j=1}^{v}w_ j(x_ j+y_ j)} $

CANBERRA

Canberra metric coefficient. See Sneath and Sokal (1973, pp. 125–126) $ d_{19}(x,y) = \sum _{j=1}^{v}{\frac{{w_ j|x_ j-y_ j|}}{w_ j(x_ j+y_ j)}} $

COSINE

cosine coefficient $ s_{20}(x,y) = \frac{\sum _{j=1}^{v}w_ j(x_{j}y_ j)}{\sqrt {\sum _{j=1}^{v}w_ j{x_ j}^2\sum _{j=1}^{v}w_ j{y_ j}^2}}$

DOT

dot (inner) product coefficient $ s_{21}(x,y) = [\sum _{j=1}^{v}w_ j(x_{j}y_ j)]/ \sum _{j=1}^{v}w_ j $

OVERLAP

sum of the minimum values $ s_{22}(x,y) = \sum _{j=1}^{v}w_ j[\min (x_ j,y_ j)]$

DOVERLAP

maximum of the sum of the x and the sum of y minus overlap $ d_{23}(x,y) = \max (\sum _{j=1}^{v}w_{j}x_ j,\sum _{j=1}^{v}w_{j}y_ j)-s_{22}(x,y)$

CHISQ

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

 

Variable

Row

Observation

Var 1

Var 2

…

Var v

Sum

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

\begin{eqnarray*} r_ x & =& \textstyle \sum _{j=1}^{v}w_ jx_ j \\ r_ y & =& \textstyle \sum _{j=1}^{v}w_ jy_ j\\ c_ j & =& w_ j(x_ j+y_ j)\\ T & =& r_ x+r_ y=\textstyle \sum _{j=1}^{v}c_ j \end{eqnarray*}

The chi-square measure is computed as follows:

$ d_{24}(x,y) = (\sum _{j=1}^ v{\frac{(w_ jx_ j-E(x_ j))^2}{E(x_ j)}}+ \sum _{j=1}^ v{\frac{(w_ jy_ j-E(y_ j))^2}{E(y_ j)}}) W / (\sum _{j=1}^{v}w_ j) $

where for j= 1, 2, …, v

\begin{eqnarray*} E(x_ j) & =& r_ xc_ j/T \\ E(y_ j) & =& r_ yc_ j/T \end{eqnarray*}
CHI

square root of chi-square $ d_{25}(x,y) = \sqrt {d_{23}(x,y)} $

PHISQ

phi-square This is the CHISQ dissimilarity normalized by the sum of weights$ d_{26}(x,y) = d_{24}(x,y) / (\sum _{j=1}^{v}w_ j) $

PHI

square root of phi-square $ d_{27}(x,y) = \sqrt {d_{25}(x,y)} $

Methods That Accept Symmetric Nominal Variables

The following notation is used for computing $d_{28}(x,y)$ to $s_{35}(x,y)$. 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 $ w_ j = 0 \mbox{, if either } x_ j \mbox{ or } y_ j \mbox{ is missing.} $

M

nonmissing matches

$M= \sum _{j=1}^{v}w_ j\delta _{x,y}^{j}$, where

\begin{eqnarray*} \delta _{x,y}^{j}& =& 1 \mr{,\ if \ } x_ j = y_ j \\ \delta _{x,y}^{j}& =& 0 \mr{,\ otherwise} \end{eqnarray*}
X

nonmissing mismatches

$X= \sum _{j=1}^{v}w_ j\delta _{x,y}^{j}$, where

\begin{eqnarray*} \delta _{x,y}^{j}& =& 1 \mr{,\ if \ } x_ j \neq y_ j \\ \delta _{x,y}^{j}& =& 0 \mr{, \ otherwise} \end{eqnarray*}
N

total nonmissing pairs

$N= \sum _{j=1}^{v}w_ j$

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 $ d_{30}(x,y) = \sqrt {1-M/N} = \sqrt {(X/N)} $

DSQMATCH

simple matching coefficient transformed to squared Euclidean distance $ d_{31}(x,y) = 1-M/N = X/N $

HAMANN

Hamann coefficient $ s_{32}(x,y) = (M-X) / N $

RT

Roger and Tanimoto $ s_{33}(x,y) = M / (M+2X) $

SS1

Sokal and Sneath 1 $ s_{34}(x,y) = 2M / (2M+X) $

SS3

Sokal and Sneath 3. The coefficient between an observation and itself is always indeterminate (missing) since there is no mismatch. $ s_{35}(x,y) = M / X $

The following notation is used for computing $s_{36}(x,y)$ to $ d_{41}(x,y)$. 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 $ w_ j=0 \mr{, if\ either\ } x_ j \mr{\ or\ } y_ j \mr{\ is\ missing.} $

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.

X

mismatches with at least one present

$X= \sum _{j=1}^{v}w_ j\delta _{x,y}^{j}$, where

\begin{eqnarray*} \delta _{x,y}^{j}& =& 1 \mr{,\ if \ } x_ j \neq y_ j \mr{\ and\ not\ both \ } x_ j \mr{\ and \ } y_ j \mr{\ are\ absent} \\ \delta _{x,y}^{j}& =& 0 \mr{,\ otherwise} \end{eqnarray*}
PM

present matches

$\mathit{PM}= \sum _{j=1}^{v}w_ j\delta _{x,y}^{j}$, where

\begin{eqnarray*} \delta _{x,y}^{j}& =& 1 \mr{,\ if \ } x_ j = y_ j \mr{\ and\ both \ } x_ j \mr{\ and \ } y_ j \mr{\ are\ present} \\ \delta _{x,y}^{j}& = & 0 \mr{,\ otherwise} \end{eqnarray*}
PX

present mismatches

$\mathit{PX}= \sum _{j=1}^{v}w_ j\delta _{x,y}^{j}$, where

\begin{eqnarray*} \delta _{x,y}^{j}& =& 1 \mr{,\ if \ } x_ j \neq y_ j \mr{\ and\ both \ } x_ j \mr{\ and \ } y_ j \mr{\ are\ present} \\ \delta _{x,y}^{j}& = & 0 \mr{,\ otherwise} \end{eqnarray*}
PP

both present = PM + PX

P

at least one present = PM + X

PAX

present-absent mismatches

$\mathit{PAX}= \sum _{j=1}^{v}w_ j\delta _{x,y}^{j}$, where

\begin{eqnarray*} \delta _{x,y}^{j}& =& 1 \mr{,\ if \ } x_ j \neq y_ j \mr{\ and\ either\ } x_ j \mr{\ is\ present\ and \ } y_ j \mr{\ is\ absent\ or}\\ & & x_ j \mr{\ is\ absent\ and \ } y_ j \mr{\ is\ present}\\ \delta _{x,y}^{j}& =& 0 \mr{\ otherwise} \end{eqnarray*}
N

total nonmissing pairs

$N= \sum _{j=1}^{v}w_ j$

Methods That Accept 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.

$ s_{36}(x,y) = s_{16}(x,y) + \mathit{PM} / P $

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.

$ d_{37}(x,y) = d_{17}(x,y) + X / P $

Methods That Accept Asymmetric Nominal Variables

DICE

Dice coefficient or Czekanowski/Sorensen similarity coefficient $ s_{38}(x,y) = 2\mathit{PM} / (P + \mathit{PM}) $

RR

Russell and Rao. This is the binary equivalent of the dot product coefficient. $ s_{39}(x,y) = \mathit{PM} / N $

BLWNM | BRAYCURTIS

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

$ d_{40}(x,y) = X / (\mathit{PAX}+2\mathit{PP}) $

K1

Kulcynski 1. The coefficient between an observation and itself is always indeterminate (missing) since there is no mismatch. $ d_{41}(x,y) = \mathit{PM} / X $